This question seems to invite speculation as a answer, but I'll try to give a few answers. Mods, if it gets too close to speculation please inform me:
Some countries did take advantage of the chaos of the ACW. France, for instance, installed a friendly regime in Mexico during the ACW. After the war ended the US began pressuring France to withdraw it's troops, and France did(which led to the collapse of the regime there).
While there's popular perception that the Confederates were "better" fighters than Union troops, it's worth mentioning that from early on the Union was advancing and winning regularly everywhere but Virginia, where the "best" Confederate officers were and they had closer parity in numbers. A string of victories and a sense of "progress" meant that, perhaps, the American army wouldn't be as much of a walkover as some in Europe would assume.
The US was(and is) enormous with few major urban areas that carried the rest of the country. While one could argue that London or Paris were "the" cities that carried outsized political and economic weight to the rest of their countries, the US in the 1860s didn't quite have similar cities. The capture of the New Orleans(the largest, wealthiest city in the Confederacy) in 1862 showed that the American(Confederate and Union) economies and political structure could absorb losses of major urban areas and keep on fighting, thanks in part to the dispersed population.
The US is big. Really, really big. Any invasion would require enormous numbers of troops. The numbers used by the Confederate and Union armies meant that the era of 50,000 or so total soldiers for all of North America(War of 1812) were long gone. Not saying no other country could scrape together the numbers, but it leads into my next point:
The European countries had a lot of competitors back in Europe, and their own problems. France just ended a war with the Austrians. The Italian peninsula was in the throes of it's own war(s) to determine the future of Italy. Stripping out troops and sending overseas to right a industrial war far from the base of supply while worrying about problems close to home would not be high on the "to-do" list of the French or British.
More money to be made selling to the US Government and(to a lesser degree) Confederate government. Going to war wouldn't get them any richer and invited considerable risk.