One of the more famous quotes comes from William Howard Russell, writing for the Times of London: " So short-lived has been the American Union, that men who saw its rise may live to see its fall."
Russell wrote those words after First Bull Run, and they've appeared in Shelby Foote's works, Ken Burns' The Civil War and all over the place since. If the world had any third-party expert on warfare, it was Russell. He was a journalist considered one of the world's first war correspondents. He cut his teeth in the First Schleswig War, went on to report on the Crimean War, then was in Washington, D.C. from 1861 to 1863.
After that, he went on to cover the Franco-Prussian War, dying in 1907.
The Times maintained a seemingly sympathetic view toward the Confederacy and the South. As Louis Maseur, author of The Civil War: A Concise History writes, " The British upper classes tended to identify with the aristocratic ambitions of Southerners, the textile industry depended on cotton imports, and London merchants thrived on exports to the United States, south as well as north."
Russell wrote his first letter to the Times in March 1861. It was published in April and appeared in the United States in May. Russell concluded, “from all I have seen and heard my belief is that the Southern States have gone from the Union, if not for ever, at least for such time as will secure for their Government an absolute independence till it be terminated by war.”
I believe it's Adam Goodheart who argues in 1861: A Social History Of The Civil War, that we should remember how fresh and unproven the American experiment in self-government was. If the Southern states were allowed to leave the Union, what would stop New York or Massachusetts from leaving a few years hence over some perceived slight or economic imbalance? The idea of Union was not as remote and abstract an idea to fight for in the 1860s as it seems to us today, after a century of thinking of the US as a single nation tightly bound together economically.
We know how the experiment turned out—with a strong continental nation—but at the time there was no such certainty.
Abroad, certainly. Close to home, the political leaders of British North America directly drew from Civil War when designing their plans for the Dominion of Canada. They thought that "mob" democracy and a weak federal system were a terrible idea. Such expressions are ubiquitous from people like John A. MacDonald his contemporaries. See Creighton, The Old Chieftan, or P.B. Waite, The Life and Times of Confederation, or Mayer, Dixie and the Dominion
According to what I've read, there was quite a large amount of debate abroad as to the true causes behind the Civil War consuming the "Grand Republic." Liberals, broadly speaking, in Europe were dismayed to see the Union dissolving into Civil War. Many aristocrats, although again speaking broadly here, were overjoyed.
The Civil War led to a temporary cessation of the Monroe Doctrine, despite Seward and Lincoln's foreign policy stating they'd go to war with any nation interfering in America's affairs, which allowed for European interference in the Western hemisphere again. For example the French attempts to build a "Latin Empire" beginning in Mexico and the Spanish reconquest of Santo Domingo.
European liberals, among them Giuseppe Garibaldi, were frustrated by the North's insistence on the constitutional impossibility of the South's rebellion rather than what many clearly saw as the central issue, the institution of slavery.
Autocratic and aristocratic governments saw the perils of mob rule and the vast slaughter of the Civil War as inevitably stemming from democratic rule. This war, along with the examples of repeated civil wars in South America, proved to them that America was not immune and that republics were inherently destined to dissolve into conflict and barbarism.
In the Confederacy, Davis and other leading politicians, argued that Jefferson and the other Founding Fathers were incorrect in their views on slavery and other principles. In a somewhat paradoxical fashion (as I recall also an almost deification of Washington and others, although I could be misremembering), leading members of the Confederate aristocracy denounced the original American Revolution. Some of the Southern aristocrats even expressed a desire to reunify with the British Crown or other autocratic nations like France and Spain.
At issue in the American Civil War, beyond the immediate question of slavery, was the divide between liberalism and autocracy. All across the European-American world, many watched to see how the chips would fall and fierce debates over the central issues of the war, over which side should be supported, consumed the public and ultimately the foreign policy directions of many nations/states.
Sources: The Cause of All Nations: An International History of the American Civil War by Don H. Doyle, years of random readings and half-remembered college lectures.
I'd like to apply a modest corrective here. It's a very common sentiment to view the history of Europe as a long continuous stretch of cohesive governments and nations, especially up through the 19th century, but this is an erroneous view.
If you look at the history of Europe at 1860 you see a recent history of young nations and of tumult. France, for example, had been cycling between Monarchy and republic throughout the early 19th century and its then current government was only a few decades old. Britain, meanwhile, had experienced a tumultuous revolution a century prior to the American revolution and had seen dramatic changes in the nature of its government, especially through the 1830s and '40s. Italy was a constantly changing hodgepodge of states, which proceeded through stages of unification through the mid 19th century, much as Germany and Prussia did. The revolutions of 1848 had transformed many of the governments of Europe and led to a comparatively new political landscape.
Additionally, it's important to remember the great political struggles that were going on at the time, especially in Europe, and view the opinions of people from that time in that light. Remember that this was a time of great and recent turmoil throughout the world and especially in Europe. There were many revolutions, some more successful than others, many wide ranging wars. And there was a major debate and political battle focused on the proper forms of governance, law, ordering of society, etc. Remember that this was the time when The Communist Manifesto was fresh. When socialists, anarchists, "liberals", radicals and so forth were first gaining major political power and challenging established power structures. It was a time of great tumult and more so of great political debate. As consequential was the debate over slavery and so forth in the US, equally consequential debates over the rights of man, the proper foundations of government, etc. were happening around the world. How people viewed the civil war and the political debate in the US would almost certainly be substantially influenced by their personal leanings towards democratic governance, radical equality, or conservative ideals or the justifications for aristocracy.
There have been a lot of comments on foreign opinions so far, but none on domestic. I think we can find an answer in the Gettysburg Address. Lincoln wrote "Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure." He then goes on to say "we here highly resolve that these dead shall not have died in vain -- that this nation, under God, shall have a new birth of freedom -- and that government of the people, by the people, for the people, shall not perish from the earth." I think Lincoln is very clearly stating here that the Civil War, at least to him and likely to many others, was a direct test of your question. We must remember that, while the Emancipation Proclamation had been released at the beginning of 1863, the same year as Gettysburg, the original northern justification for the war, and one that did not disappear after the Emancipation Proclamation, was to preserve the Union. For many people in the north, the war would be an active attempt to prove that the United States would not fail.
I just finished a really excellent book on this subject, The Cause of All nations by Don Doyle. He traces different European and other foreign reactions to the war and how they changed from 1861-1865.
He makes the case throughout that European conservatives (British aristocrats, the French emperor, right-wing Catholics) took the outbreak of the war as confirmation that democratic governments could not last. They supported the Confederacy and took advantage of the United States' troubles to re-colonize areas of the Americas.
European liberals and republicans, by contrast, saw the war as a test for democracy and supported the Union, often as a way of obliquely criticizing their own governments. Here's a quote from the British reformer John Bright that explains how the two sides differed:
Privilege thinks it has a great interest in this contest, and every morning, with blatant voice, it comes into your streets and curses the American Repubhc. Privilege has beheld an afflicting spectacle for many years past. It has beheld thirty millions of men, happy and prosperous, without emperor, without king, without the surrouiidings of a court, without nobles, except such as are made by eminence in intellect and virtue, without State bishops and State priests, without great armies and great navies, without great debt and without great taxes. Privilege has shuddered at what might happen to old Europe if this grand experiment should succeed.