I think one well known recent example (just within the bounds of this sub’s rules), about which a lot has been discussed and written, and even already mythologised, is late Apartheid South Africa.

A little bit of basic background: in 1910 South Africa unified from British colonies and ‘Boer’ (or rather Afrikaans, or essentially Dutch South African) republics founded in the interior, in the wake of the 1899-1902 (‘Second’) Boer War, where the British Empire took over the latter. From the start (and in its predecessor polities) there was racial discrimination against the black majority and other groups (largely self-described ‘Coloured’ people and Indians), with restrictions on voting, work, living conditions etc. But in 1948 an Afrikaner nationalist government took power and began to institute a series of laws that formalised race and segregation in every avenue of life as the fundamental system by which the country operated, eventually consigning black people to so-called tribal ‘homelands’ or derisively named ‘Bantustans’, essentially reservations taking up 15% of the country, with the pretence that they were independent nation states, declaring all black people to be citizens of these states and with the goal of removing them from the remaining 85% of the country to these small areas.

Black people did not take this lying down, and several movements formed or resolved to fight these oppressive measures. The most famous was the African National Congress, which was already in existence from 1912, and eventually led by Nelson Mandela, who was arrested for sabotage and imprisoned for 27 years. Others included the Pan Africanist Congress and Steve Biko’s Black Consciousness Movement. There were also liberal and left-wing movements among the white population, and other groups. The first two had armed or military/paramilitary wings: Nelson Mandela’s uMkhonto we Sizwe (MK) and the PAC’s Azanian People’s Liberation Army (APLA - named after the Ancient Greek references to a land called ‘Azania’, taken to be in sub-Saharan Africa). There was conflict: these and others conducted resistance campaigns that involved bombings, targeted shootings, mass protests, interaction with international organisations mostly successfully pushing for sanctions against South Africa, and proxy conflicts.

So there was, of course, violence. Unsurprisingly, these groups were banned and resistance generally fought with arrests, torture, assassinations/murders, repression and, on occasion, massacre.

There were also proxy wars of sorts: when South Africa’s neighbours were either very small or even surrounded (Swaziland, Botswana, Lesotho), or white-led (two Portuguese colonies, now Angola and Mozambique, and white-led Rhodesia, now Zimbabwe). What is now Namibia was in effect a South African colony or even province - a League of Nations ‘mandate’ they had inherited at the end of WW1 as the British Empire’s ‘custodian’ over the former German colony.

But when the Portuguese colonial empire collapsed in 1974-5, the South African government found for the first time that it had two left wing, anti-colonial and black-led governments on its doorstep (Mozambique and Angola - the latter neighbouring South West Africa), which were used as bases by the black resistance movements, with those governments’ support. A proxy war was fought in Angola and SW Africa, which pitted the South African government against Angola’s MPLA, its allies in what is now Namibia (SWAPO), Cuban forces sent by Castro, and black South African resistance fighters. This war did not go well for the South African government, which eventually withdrew and by 1990 Namibia was given independence as well.

As this continued, other factors also played a role: the white population’s share of the general population declined, as many left and as birth rate was outpaced - in the earlier twentieth century it had been as high as 22%, but now it is closer to 6-7%. There was international pressure and increasing internal pressure not only from these movements but among the general white electorate, which was leaning more and more against the Apartheid system as all the impacts were felt, the general global norm turned against segregation and racism, and a new generation grew up. It became clear to the sixth leader of the Apartheid government, De Klerk, that the system was untenable and that in order to avoid a civil war, negotiations were essential, and over a series of - by both accounts, not always agreeable - meetings with Mandela and others, he agreed to release Mandela in 1990 and propose legislation to abolish the Apartheid laws. Despite resistance from hard-liners he held a (whites-only…) referendum on his decision in 1992, and the result was majority approval. In 1994 a pan-racial election was held and Nelson Mandela’s ANC won in a landslide. Nelson Mandela’s first speeches and his messages after that were fundamentally based on the idea that the transition had to be as peaceful as possible, and he took several extraordinary steps to keep the white electorate at ease, including appearing at Afrikaans and Anglo-South African cultural events, adding snippets of Afrikaans in speeches (by many black South Africans English was seen as a ‘neutral’ language, while Afrikaans was seen as the ‘language of the oppressor’), and explicitly emphasising non-violence. The constitution was rewritten (in great part by Cyril Ramaphosa, an ANC leader who is now the current president) with racial equality and anti-discrimination laws emphasised strongly.

Desmond Tutu, the black Anglican Archbishop of Cape Town, whose appointment in the 1980s had caused a stir, was another famous proponent of peaceful transition, to the point that he chaired a Truth and Reconciliation Commission, which held forums where people who had committed violence during the Apartheid era - on whichever side - could come forward with immunity and confess and discuss their crimes with the victims or their families. Though based on a spirit of forgiveness and seen by many as helping to move the country forward, it has not been without criticism, as many people felt that those who committed atrocities got off lightly simply by tying their atrocities to politics, even repressive politics. It has, however, been imitated elsewhere, including in Liberia. There is literature discussing the effects this had or did not have on the peaceful transition, which I will try to find.

Many factors avoided civil war: in addition to the above it has to be mentioned that despite its loss in Angola, the South African military was very well equipped and, finding itself isolated, even developed nuclear weapons (on the democratic transition these were dismantled - South Africa being the only country ever to voluntarily rid itself of all its own nuclear weapons). This added to the certainty that a full-blown conflict at home was something both sides wanted to avoid. There are also questions about what the most important forces were that ended Apartheid: there is a great deal of literature about the resistance, the most obvious, but the question of whether sanctions helped, did little, or even hindered the process is another thorny one with substantial discussion in the literature.

There was still violence, however: again, not anything like a civil war, but despite condemnation by their respective leaderships, many supporters of the IFP (a largely Zulu party in the east of the country whose leader - a member of the Zulu royal family - had emphasised ending Apartheid through talks and working within the political system of its day, and had led the Zulu ‘bantustan’), and the ANC (a party more dominant among the other black ethnic groups, like the Xhosa and Sotho) had violent confrontations in which many died, including a proxy ‘taxi war’. There is also the issue of a ‘third force’ formed by disgruntled members of the former white government and military, and possible false flag missions they conducted to instigate further ANC-IFP violence. This is a controversial issue, given that much of it was clandestine and the senior leadership of none of the three parties explicitly acknowledged any involvement.

In many ways this is similar to another process that took place at around the same time: the dissolution of the Soviet Union. In both cases the leader who began to acquiesce to reforms had actually had a reputation as a loyal ideologue while climbing the ranks but softened their position for pragmatic reasons once in power; there were hard-liners in the government who refused to cooperate (though there was no South African equivalent of the 1991 coup attempt there); and the first leader who founded the more democratic government after had a reputation for a ‘courageous stand’ and opposition during at least the last years of the regime, but also called for peaceful transition (though of course in Yeltsin’s case, he had been a member of the government not too long before this role); and in both cases some sort of union with at least one other state was dissolved as a consequence. One bizarre difference is that despite Yeltsin’s political origins as a communist, he banned the Communist party once in power. On the other hand, Mandela not only did not ban the Nationalist Party as they had banned his organisation, he formed a Government of National Unity with them, and De Klerk became the deputy Vice President, nominally third in command. (This took an even more bizarre twist when, after a transition from a white nationalist party to having a largely ‘Coloured’ base, and various political dealings, the National Party, once the party of Apartheid, was officially absorbed by the ANC, the once black nationalist party that they had banned as a terrorist organisation. But this goes beyond the period of concern.)

This is a massive topic so I could not go into as much depth as it deserves. I will try to add more details and sources when I get time.

I believe that Belgium in the years that followed WW2 fits your example of a country that was on the brink of civil war but where a full-blown conflict ended up being avoided.

In May 1940, during the battle of Belgium, the Germans forces advanced very quickly through Belgium’s territory. While in WW1 king Albert I fled with the government to France to establish a government in exile, this time, his son and successor king Leopold III decided to stay in Belgium and to surrender, while the government fled to France (and later to London). This move was immensely unpopular and seen as an act of treason, both for the Belgian population and government. Prime Minister Hubert Pierlot invoked article 93 of the Belgian constitution, and Leopold III was officially declared unable to rule.

His actions during the occupation further tarnished his image. From 1940 to 1944, Leopold III was placed under house arrest in the castle of Laeken. This was in stark contrast with the previous king, Albert I, who regularly visited soldiers in the trenches. On top of that, Leopold met Hitler in Berchtesgaden. In 1941, Leopold also married Lilianne Baels, a commoner. This marriage proved to be deeply unpopular, because the Belgian population was still attached to his previous wife, queen Astrid, who died in a car accident in 1936. At the time, many Belgian men were still prisoners of war in German POW camps and separated from their wives, which led to even more frustration towards the privileges that the king enjoyed.

When the allies approached Belgium in 1944, the king was deported to Germany and later to Austria. Because the location of the king was unknown to the Belgian government in exile, they decided to appoint Charles, the younger brother of Leopold III, as regent. Unlike Leopold, he had not been taken prisoner to Germany, and he was next in the line of succession, after Leopold’s two sons who were still teenagers at the time.

After WWII had ended, the declaration of the king’s inability to rule had not been revoked, which led to the crisis known as the “Question Royale”, that could have possibly turned into a civil war. The question was the following : Should Leopold III be allowed to return to Belgium? And if so, should he remain the king”?

In the following years, Belgian society was heavily divided between Leopoldists, who favoured the king’s return, and anti-Leopoldists, who were against his return. The question divided not only the people, but also the trade unions, the newspapers... The political parties also took sides on the issue : the Liberal Party, the Socialist Party and the Communist Party wanted the king to abdicate, while the Social-Christian Party favoured the king’s return. In 1949, the Social-Christian Party under Gaston Eyskens campaigned on a pro-Leopold platform and won the election.

The newly elected government agreed to a national referendum on the return of the king, which was set to take place in 1950. In the end, 57% of the Belgian population voted for the king to return, but the referendum revealed fracture lines between the two communities, as Wallonia, with a strong socialist presence, was mostly hostile to the return of Leopold (58%), while a mainly Christian Flanders was in favour (72%).

His return in July 1950 caused a massive and violent general strike, and street fighting between the two camps occurred all over the country, particularly in Wallonia and Brussels, and to a lesser extent in Flanders, where the port of Antwerp was also paralysed. During the strikes, the gendarmerie killed 4 protesters. Some Walloon politicians even flaunted the idea of the separation of the region from Belgium.

A month after his return, Leopold III realised that his position was untenable. The Prime Minister, Jean Duvieusart, visited him and pressured him to abdicate rapidly. A day later, Leopold III officially abdicated in favor of his son, Baudouin. On the same day Baudouin swore allegiance to the constitution, Julien Lahaut, the leader of the Communist party, was assassinated, most likely by Leopoldist militias. This event is considered as the end of the crisis.

While the abdication of the king calmed the tensions and a full-blown conflict was avoided, the crisis left a mark on Belgian society. Flanders, which had voted in favour of the return of the king, saw his abdication as a diktat from Wallonia, and as a result the Flemish Movement grew in influence. The Question Royale is seen as one the causes of the transformation of Belgium from a unitary State to the federal State it is today.

One of the finest ancient examples of this - civil war prevented by extraordinary measures - is the story of the Athenian sage, poet and reformer Solon. But this is a story of imperfect conflict resolution with many caveats and footnotes.

When Solon was elected to the highest magistracy of Athens in 594 BC, the community seemed hopelessly divided. On one side were the rich: a closed aristocracy calling itself Eupatridai (the well-born), who owned the land, monopolised political power, and competed with each other in extravagant displays of wealth. On the other side were the poor, who found themselves crushed by rents levied by the rich, and who were increasingly defaulting on their debts and falling into debt bondage. The poor demanded debt relief and redistribution of land; the rich, afraid to lose their status, refused. Both sides took recourse to intimidation, violence and theft. Solon himself vividly describes what was about to happen:

This [Justice's vengeance] is now coming upon the whole city as an inescapable wound and the city has quickly approached wretched slavery, which arouses civil strife and slumbering war, the loss for many of their lovely youth (...) And so the public evil comes home to each man and the courtyard gates no longer have the will to hold it back, but it leaps over the high barrier and assuredly finds him out, even if he takes refuge in an innermost corner of his room. (fr. 4)

In the midst of this crisis, likely because of Solon's politically engaged poetry and his track record of serving the public good, the leaders of both sides requested him to act as the arbiter of the city's ills, and to introduce whatever measures he thought might prevent all-out civil war.

Solon understood that simply giving one side what it wanted and surpressing the other was certain to lead to violence. Confiscating the land of the rich and redistributing it would create a band of furious Eupatridai with vast movabe wealth, well-cultivated foreign contacts, and nothing to lose. On the other hand, keeping things as they were to gratify the ruling class would either lead to a popular uprising or to the dissolution of the community as the remaining common people moved away or were sold into slavery. Several of the surviving fragments of his poetry show that he thought neither outcome could fix Athens, and that each side had to be protected from the other:

I stood with a mighty shield cast round both sides and did not allow either to have an unjust victory. (fr. 5)

If I had been willing to do what was then pleasing to their opponents, and in turn whatever the others planned for them, this city would have been bereft of many men. For that reason I set up a defence on every side and turned about like a wolf among a pack of dogs. (fr. 36)

I stood in the no-man's-land of spears like a boundary stone. (fr. 37)

Or at least, that is what Solon boasted he had done. His measures seem to have prevented civil war at Athens for the time being, that much is true - but what he did to achieve this has been a subject of debate for centuries. Our evidence is extremely thin and incredibly tendentious; some of it was written by Solon himself to justify and glorify his actions, while the rest comes from later Athenian sources who used Solon to make a point about a supposedly glorious past when Athens was more harmonious. It is enormously difficult to try to reconstruct how exactly Solon resolved the crisis.

His reforms of the political system, including the introduction of a new roster of property classes with associated political rights, have traditionally been seen as an effort to give a rising middle class access to power in the city. But more recent research has shown that Solon's property classes totally disregard anything that we would consider a middle class; 3 of the 4 classes he introduced are effectively different tiers of leisure-class landowner. Every small farmer, shopkeeper, merchant, day labourer - the great majority of the population - is lumped together in the lowest class, dismissively called thetes ("wage-earners") and granted barely any legal or political rights of any kind. Solon was very explicit that he arranged things this way on purpose:

I have given the masses as much privilege as is sufficient, neither taking away from their honour nor adding to it. (fr. 5)

Solon may have abolished debt bondage and so liberated many poor Athenians from slavery, by which he removed the immediate cause of their distress. But he does not seem to have done much else to protect them from the predations of the rich. Later claims that Solon was the first ancestor of Athenian democracy ring hollow against passages from his own poetry:

And in this way the masses would best follow their leaders, if they are neither given too much freedom nor subjected to too much restraint. (fr. 6)

Obey rulers, however right or wrong. (fr. 30)

It is clear that Solon was much more concerned with managing relations within the minority of wealthy landowners. His system of property classes entails the first definition of political privileges in Athens - basically, how rich you had to be to get on which ride. Various legal reforms were aimed at preventing the monopolisation of power and increased the accountability of magistrates. This would all eventually have a democratising effect, but it is not clear at all that it was intended to do so; most of these reforms were more likely intended to give the elite the means to police itself. The intention was to prevent at Athens what was commonly happening all around the Greek world in this period: the seizure of absolute power by that member of the wealthy elite who "won" the game of elite competition and gained enough status to dominate all his rivals. Solon's preoccupation with tyranny is extremely clear from his poems, in which he brags repeatedly that he didn't seize the tyranny himself, though he could have, and others would have done so if they were in his shoes:

If I spared my homeland and did not grasp tyranny and brute force, bringing stain and disgrace on my reputation, I am not ashamed. For I think that in this way I shall be more able to outstrip everyone. (fr. 32)

Yet, if the purpose of Solon's reforms was to prevent the competition of the rich from leading to tyranny at Athens, they failed. As the author of the Aristotelian Constitution of the Athenians pointedly notes, the years following Solon's reform and departure from Athens (supposedly to keep himself from caving to people lobbying for further reforms) saw several irregularities in the higher magistracies, either posts remaining vacant or people refusing to leave when their time was up. A few decades later, in 561 BC, the wealthy Peisistratos seized the tyranny. For the next half century or so (though it took a while for the tyranny to become firmly established), Athens was under the rule of Peisistratos and his sons, whose rule was to become ever more violent and oppressive. The final result was assassination, foreign intervention, mass exile, and civil war - the very things Solon had been trying to prevent. The Athenian political system did not become stable until the further reforms of Kleisthenes in 507 BC.

One of the details that's compelling about your question is how we interpret the idea of a civil war and who we visualize when we think about the combatants. At first blush, it appears outside the scope of what you're asking, but I would offer that American education offers multiple examples of extreme polarization where a full scale civil war was narrowly avoided.

The most well-known education "war" is probably the math wars (which warrants its own Wikipedia entry) but the idea of "war" is a common way of framing events and conflict in American education history. One of my favorite books is The Teacher Wars about the feminization of the profession and the impact of systematic sexism on the field, and then there's the Classroom Wars about sex ed in California schools. We're currently living through what is easily the 5th or 6th iteration of both the history wars and the reading wars. And then there's the OG: The Testing Wars about the push-pull for large-scale standardized testing in Boston in the 1840s that came very close to large-scale violence (there were instances of smaller scaler violence as there was in all of these wars I mentioned.)

To be sure, a fair amount of this is purely rhetorical - education history is fraught with battles of words - but the framing of a "civil war" works well in these instances even with start with the premise that the two sides in the war are more alike than not; that they're connected by a number of commonalities but one particular issue creates a fracture that threatens large scale violence. (Historian Benjamin Justice refers to this as "The warfare thesis" in his book, The War that Wasn’t: Religious Conflict and Compromise in the Common Schools of New York State, 1865–190) Another reason I wanted to offer an example from education history is it allows us to think about the history of women and children in a slightly different way. While women and children have been a part of every war since time immemorial and are impacted by war in a multitude of ways, there are "wars" that happen off designated battlefields that have far-reaching consequences on women and children. And the almost-civil war between Protestants and Catholics over specifics related to American public education is a prime example of that.

A few big picture things to set the stage. There is no national education system in the United States which means every state developed its own system and has its own structure. However, despite these different histories, certain touchstones can be seen in schools in all 50 states. Described by education historians as the "grammar of schooling", they're the things that make it clear you're in a school. Things like bright colors, apples, bulletin boards, etc. Each piece of "grammar" has its own history and its own origin story, but a number of details - including coeducation as the norm and calling teachers by a gendered title and their last name - have their roots in Protestantism. Most, if not all, of the early advocates of a common - later public - school structure beginning in the 1830s were Protestants, to one degree or another. In a practical sense, this meant that early primers and texts that children used in schools were likely to be religious texts, based in Protestant tradition.

This was non-controversial for most white children and their parents until there was a sense that Catholic immigrants were reaching a critical mass or until Catholic men moved into positions of power, which varied dramatically by city, especially after the American Civil War. As an example, school boards in and around Boston shifted back and forth between Catholic and Protestant control while those in Philadelphia remained firmly under Protestant control. Likely the first shot (as it were) in this particular skirmish happened in Cincinnati in 1869 when the Protestant-controlled school board agreed to remove certain Protestant texts from the curriculum if the Catholic diocese moved control of their schools out from the diocese to under the school board. But, at the 11th hour, a Catholic newspaper published an article about the agreement and the diocese backed out. (These events were known as the "Cincinnati Bible War” and would shape early legal theory around the separation of church and state in public schools.) Meanwhile, the NYS legislature and the NYC school board, in particular, were up to all sorts of shenanigans involving public funding of Catholic schools as a way to leverage control over Catholics in the city. (Diane Ravitch's "The Great School Wars: New York City, 1805-1973—A History of the Public Schools as Battlefield of Social Change" (1974) is the most comprehensive text about these various battles.)

By the 1870s, being openly and actively anti-Catholic was seen as a viable political move among politicians, including among school board members or schoolmen, men who lead schools and districts. In many places, Catholic dioceses responded by denying parents who enrolled their children in public schools, rather than Catholic schools, the sacrament. From Justice's book:

After a disastrous Republican Party defeat in 1874, President Grant played to anti-Catholic sentiment in an 1875 speech warning that if the nation should again go to war, the new Mason-Dixon line would be drawn at the common schoolhouse door, and sectarian (Catholic) influence would be the new enemy. Radical Republicans in Congress sponsored a constitutional amendment banning public money to church-controlled schools and sectarian instruction in public schools. The amendment rocketed through the House (180 yeas, 7 nays, and 98 abstentions), later falling short in the Senate.

In your question, you asked about a solution and the "civil war" between Protestants and Catholics as it related to education came to an end through the rise of the (mostly) well-funded, (sometimes more than others) well-attended parochial school system. By the 1950s, American schools had lost most of the vestiges of Protestantism tradition (save a few like those I mentioned previously) and had become secularized (save the whole "under God" thing in the pledge and local, mostly Southern and more Evangelical, traditions that are beyond the scope of this answer) but the scars, as it were, of the original war remain.