I will take the opposite opinion of DuxBelisarius with this question.

Just in terms of a land Army, the Union Army was larger than any European army, it had more practical combat experience, and it had a vetted and seasoned cadre of officers learned in the school of war. And thats if we exclude demobilized Confederate soldiers. While many early war Union generals are lambasted for their inability to control their armies, the same cant be said of Grant, Sherman, and Sheridan. These men had been fighting for years, and had risen to the top ranks of the Union Army based solely on their ability to successfully prosecute wars. Simply put, they were the best generals the north had to offer, and they had just been victorious. But the same can be said of many of their subordinate generals, and even the lower officers. Men who had long risen above their rank in peacetime had been removed or demoted under wartime necessity.

In addition, the tactics and doctrine of the US Army had radically evolved during the ACW. Infantry had learned how to operate on a battlefield dominated by rifled cannon and muskets, how to properly deploy skirmishers, dig rifle pits, and take enemy trench works. Emory Upton's attack during the Battle of Spotsylvania was visionary, and presaged a new kind of tactical offensive. No other Army in Europe had developed these tactics, because no other army in Europe had the experience of the Union Army. The trench battles which typified the later ACW itself was conducted on a scale completely unseen in European war. Grant's Overland Campaign placed Lee and Grant in near constant contact for months, at a time when most European armies expected to fight for a few days at once.

Finally, while the Union Army was still reliant on European muzzle loading imports for the bulk of the militia forces, its incorrect to conclude that they lacked any modern firepower at all. The Union cavalry, for example, was largely equipped with very modern repeating rifles. In addition, those weapons had seen distribution to other troops as well. While a modern breech loading rifle was still a ways off, its incorrect to say that the United States simply had no modern rifles. In fact, the American's repeating rifles were some of the most advanced in the world at that time. The Dreyes was something different altogether, but was only in Prussian service.

What-If questions are really hard to take seriously. We could design any number of crazy scenarios in which the United States could play its strengths and defeat a pan-European force. Or, just as easily, we could envision a scenario in which a Prussian Army exploited the weaknesses of the American Army to achieve a paper victory. The fact of the matter is, in 1865 the forces in the US had more combat experience than any European Army. But, by 1870 things would reverse themselves. The American armies demobilized, while the Prussian Army completed its Wars of Unification. Those wars changed the way European Armies envisioned war, and which armies were believed to be the best. In the ACW, Americans radically departed from traditional European conception of war. Tactically, logistically, and operationally Helmuth von Moltke (the Elder) departed just as radically from previously established European norms. So its really hard to compare the two, or anything that lays in between.