Partially. The Congress System devised at Vienna was incredibly stable. Some historians who would back up Lavrov on this would be Henry Kissinger, AJP Taylor, and Philip Bobbitt, among many, many others. However, they wouldn't go so far as to pin WWI, or even the dissolution of international order, on the Crimean War.

The reasons for Vienna's success were several:

1. It institutionalized dialogue between the European Powers. A triennial congress was to be held where the heads of state, or their delegates would meet to discuss the contemporary state of Europe and create a cohesive plan for peace in the face of it. For example, during the Greek Civil War, rather than dragging the Balkans into a wider war with Russia and the Western Powers (like 1914), the Great Powers discussed how it ought to be resolved, leading to a contained conflict rather than general European one.

2. It came after world war levels of destruction. The Revolutionary Wars (1792-1815) had destroyed the status quo of power in Europe, bringing national armies from Madrid to Moscow. Aristocracy and the feudal order had experienced an existential crisis from which they would struggle to recover. Combined with the modernization of the economy, Napoleon's discrediting of the class system on the battlefield ensured social instability and a more powerful, conscious underclass. The participants at Vienna knew their domestic situations were perilous and their recent memories were of over two decades of war. Thus they strove to create an environment of peace. Rather than exact exorbitant concessions, with the victors exploiting the losers (think Versailles), France was invited to participate in negotiations (represented by Talleyrand), the Great Powers were placated, and their borders rationalized.

3. The participants did away with "Balance of Power" politics. This was a paradigm shift. European affairs had long been one in which states expanded as their power increased, or lost territory when they weakened. This kept the arms race of rifles, artillery, General Staffs, and state bureaucracies efficient, but it also meant constant war. Stemming from my last point, after Napoleon, Europeans finally realized that war wasn't all that much fun (most likely due to the aristocrats finally having their lives and positions threatened). At Vienna, they strove to accept the newly constituted and rationalized borders and work out their small quibbles with dialogue. They strove for universally accepted equilibrium rather than a Balance of Power. In the one, countries are fairly equal in power, though one may be moreso than another, but this is not to be broken or tested. It replaced the BoP idea, whereby one's armies were for the sole purpose of testing the limits of your neighbors' power, and picking up the territorial scraps. The whole gloire thing of invading a small principality like Alsace to show Europe your dick is bigger is a real burden to everyone when that may or may not end up extirpating a generation. So they decided to try and stay away from the dick measuring contests

4. The German Confederation was sheer brilliance. It turned the Holy Roman Empire of 350 odd statelets into a confederation of 30 larger, yet still small, states. The observation was that when Germany was unified, it was capable of horrific things in Europe-the most recent example being Napoleon's reliance on Germans in his march into Russia. However, when weak, it attracted the gluttony of European rulers, notably Louis XIV or the Prussians. So the German Confederation was created to enable ease of defense whereby all thirty would fight to protect each other, but be too fractious to mount a unified foreign invasion.

Now, on to the meat of your question. What caused the instability of the second half of the nineteenth century? The Crimean War certainly didn't help anything. Britain and France wanted to keep Constantinople out of the hands of the Russians, as this would upset the European equilibrium. However, several years before that, in 1848, the new Austro-Hungarian Emperor, Franz Joseph, had had to call in the Czar's army to put down the rebellions he was facing. This deeply discredited him in the eyes of the international community, leading Western Powers to worry about Russia's creeping hegemony and Central European weakness, possibly prompting the Crimean War. After the large wave of rebellions of 1848, the Crimean War marked the first wide resort to violence in Europe since 1815. However, it was short, contained, and resulted in a satisfactory peace. I wouldn't say it destroyed the Vienna settlement, but it did end the era of amicable and multilateral diplomacy that had existed in Europe post-Napoleon.

What assuredly did destroy the Vienna settlement was the creation of Germany. This didn't simply shake the foundations of European stability, it shattered them. In this, you see the threat of German power (point 4) reified into a state with a larger population and industrial potential than any other in Europe. All of a sudden, the equilibrium of Great Powers, predicated on all being relatively equal, was dispelled. Austro-Hungary got whipped in 1866 at Koniggratz, and in 1871, Paris was seized in, what, six weeks? Germany was easily seen as the most powerful country on the Continent, with only Imperial Britain, isolated and disinterested, possibly able to challenge her.

At the end of the day, I would take issue with Lavrov's "revisionist" attitude that when other powers turned against Russia the international system fell apart. Russia overreached and got deservedly spanked by the international community. Diplomatic channels were open, yet Russia failed to exploit them, instead goading the ire of France and Britain. Russia's main role in the post-1815 settlement was to act as a bookend to Europe. To not encroach on others (what, Czar Nick you say you don't have enough territory already?) and to use its army to put down any systemic threats: a Napoleon or domestic insurrection. In reality, the main problem was domestic. Nationalism, industrialization, and modernization changed states' priorities and methods of interacting on the world stage. But that's for another time

Consulted Sources: Henry Kissinger (before criticism, please look up his academic career focusing on 19th century diplomacy, particularly the Congress of Vienna): The White Revolutionary; Diplomacy; World Order; and A World Restored: Metternich, Castlereagh, and the Problems of Peace and Diplomacy

Philip Bobbitt The Shield of Achilles

AJP Taylor The Habsburg Monarchy

Paul Kennedy's The Rise and Fall of the Great Powers

William McNeill The Pursuit of Power

Paul Johnson The Birth of the Modern

Simon Schama History of Britain

Lectures at Wesleyan University, Yale University, and Emory (from the Great Courses)

Apologies are in order. I saw this on my phone while waking up and had to respond. Editing it now to remove all the bleary-eyed weirdness that crept into this

The thing to remember about the Vienna Settlement is that not only was it an attempt to restore the antebellum status quo of Europe, it was largely done at the expense of France, and was in many respects deliberately designed to prevent her from ever being in a position to dominate the continent again. France, therefore in contrast to the other great powers, was extremely unsatisfied with what was considered an unjust treaty and proved from the moment the ink was dry determined to revise its terms. The settlement, which prohibited acts of aggression without the approval of other great powers, was applicable to Europe and Europe only. This is arguably what motivated Napoleon III to meddle in the Near East under the pretence of protecting the rights of Christians in the area. This had the intended effect of out raging Russia, who not only had imperial interests in the area, but also claimed a special right to protect Christians as well, something which the French were keenly aware of. Ultimately, the narrow self interest of the great powers of this period, and their expansionary aims, undid the settlement. When Britain was faced with the option of checking French meddling, or supprting the hugely powerful Russia (who were an increasing threat to British imperial interests in the East) there was really no choice to be made.

The Vienna Settlement was a relative success considering the deep structural instabilities inherent in European diplomacy, and the destructive force of imperial expansion common to all powers. No Great Power possessed either the foresight or the inclination to maintain the Settlement indefinitely, and it wad perhaps due only to the trauma of the Napololeonic Wars that it lasted as long as it did.

Ultimately, French revisionism, and the lack of desire on behalf of the other great powers to enforce its terms, undid the Vienna Settlement. Russia, as guilty as any great power in the period of destabilising imperial expansion, and of excessive hostility to French ambitions to return to the Great Power top table, may appear the wronged party in the Crimean War, but it did as much as any power bar France to bring it about.

Source (from memory): Bridge and Bullen, The Great Powers and the European States System.

Just? That is a matter of opinion. If, like Lavrov, it's your opinion that a just European system is one in which Russia is a great power, arguably the strongest one, you will find it just. If you view justice through something other than a Russian nationalist lens, you may very well disagree.

Stable? Nonsense. The period 1815-53 in Europe saw (in no particular order) the Spring of Nations with its associated wars and near-wars, a great power crisis over Egypt, the breakup of the Netherlands, another French invasion of Spain, Greek independence, the overthrow of the Bourbons just 15 years after their restoration, a Russo-Turkish war, 3 Polish uprisings not counting the one in 1848, Britain on the verge of breakdown, a Bonaparte restoration, liberal revolutions and dynastic civil wars in the Iberian Peninsula, the Decembrists in Russia, and Italy kept "stable" by Austrian bayonets. The Vienna settlement was a futile attempt to wind back the clock to the age before 1789. Even the notion that it lasted as far as 1853 is giving it more credit than it deserves. The only thing that's special about the Crimean War in this context is that it was the first armed conflict between the European great powers since the fall of Napoleon, but there had been more than enough "lesser" wars and great power war scares in between. I find that definition of stability to be very constraining.

The Crimean War caused World War I? They're 6 decades apart. In that period you'll find many events that were made possible (notice that I don't say "were caused") by the Crimean War and that made possible (notice that I don't say "caused") World War I. Logically, any of these events have a more sound claim to have caused World War I. The Russo-Turkish War of 1877-78, to take a non-random example - but that would require Russia and Gorchakov to be in the wrong. To go in the other direction, why doesn't Lavrov say that the Congress System caused the Crimean War? Why does he act as if 1914 developed organically out of 1853 but 1853 was an aberrant detour from the course set in 1815?

As for Napoleon III, the notion that the Crimean War betrayed an anti-Russian mindset is just another instance of the common Russian obsession with the presumed enmity of the West. Napoleon III tried to improve relations with Russia before that war was even over - which was part of the reason why, when France and Sardinia proceeded to undermine the Vienna settlement in Italy at Austria's expense, Russia reacted with a friendly neutrality. The other part was simply retribution for the Austrian attitude during the Crimean War. When Napoleon III showed himself partial to the Polish rebels in the January Uprising, Russia's retribution turned towards France. Having stood idly by while Austria was taken down a peg, it now did the same as Bismarck created the German Empire. That's the "consistent and far-sighted policy" he mentions. It's nonsensical to talk of Gorchakov's efforts to avenge the defeat in the Crimean War as if they somehow existed outside the course of European history. A line drawn between 1853 and 1914 will pass through St Petersburg, just like it will pass through Paris and Berlin and Vienna and London, many times - and several of those throughout the career of Alexander Gorchakov. There's nothing far-sighted about obsessing over past humiliations and stirring any hornet's nest you can find in search of opportunities to undo them, whether the person doing it is Lavrov's predecessor or his boss. But that discussion breaks the 20-year rule.

in the wake of the desire to marginalise Russia in European affairs ... Russia was defeated in the Crimean War of 1853-1856

Lavrov forgot to mention that Russia was responsible for starting Crimean war, since they've attacked Sinope against previously signed treaty.

From Alexander William Kinglake, 'The Invasion of the Crimea: Its Origin, and an Account of Its Progress Down to the Death of Lord Raglan, Volume 1':

This onslaught upon Sinope, and upon vessels lying in port, was an attack upon Turkish territory, and was therefore an attack which the French and English Ambassadors had been authorised to repel by calling into action the fleets of the Western Powers.