Here's a good post a from while ago about how the American Civil War combatants attempted to use Napoleonic tactics but continually failed in execution.

The commenter also talked about the Austro-Prussian War (1866) combatants and how the Prussians used breechloading rifles, percussion capped explosive shells fire breech loading artillery, and skirmishing lines to great effect against the much less adept Austrian military.

We of course cannot know how well a continental European army would have fought the ACW armies, but it seems weighted towards Europeans.

The main thing is that the beginning of the war, and the end of the war were drastically different.

At the outset of the war, the Union and Confederate armies fought like Napoleonic forces of the beginning of the century. That is, massed artillery fire, close-order ranks marching straight forward to press the attack.

Most regiments of the time weren't well equipped especially in the South. States raised plenty of regiments, but they only equipped those regiments that they kept for home defense. A Scythe of Fire by Steven Woodworth and Warren Wilkinson has some excellent insight on this dynamic.

A lot of troops were equipped with smoothbore muskets, with the 1859 Harper's Ferry rifles being highly prized, but only a few companies were equipped with them.

In terms of technology, the Union, and to a lesser extent, the Confederacy outpaced Europe by the end of the war. Repeating rifles were widespread amongst cavalry troops, artillery could fire further and more accurately (a lot of this was experience and training improvements), and most regiments had moved almost wholly away from smoothbores and were equipped with much better rifles than what was there at the start of the war.

Tactics evolved greatly. The war. by the end, was mainly a defensive one. Actions such as Cold Harbor revealed the horrors that defensive emplacements could inflict on a Napoleonic attack.

Leadership by the end of the war was astounding in how much it had changed. Most of the political appointments at the beginning of the war were gone, and a lot of the top commanders were where they were needed. Lee, Grant, Stuart, etc. were all very capable commanders who could outcompete most of what the European powers could field, especially since they hadn't burned through the majority of their political commanders like the Americans had.

Overall, I'd say the the Americans were the superior military by the end of the war, in Napoleonic terms