I was recently watching Ken Burns' 'The Civil War' and two things stuck out to me. It said when Grant took over as overall commander of the federal forces, he was in control of the largest army in the world. And that from the first time the Ironclads went into battle, every other navy in the world was obsolete.
So my question is, given its size, experience, and technological advances, were there any other nations in the world who had the strength to possibly defeat the Union army as it stood at the end of the American Civil War? Could any other navy have overcome the US Navy and its Ironclad(s)? How much more dominant was the United States to the military forces of other nations? And how long did it take the rest of the major world powers to catch up in terms of military strength?
A discussion of who would beat who is likely better at /r/HistoryWhatIf, so I won't say much there.
I will chime in and say that, while the Union Army may have been larger than any of the European armies, it's important to remember they were very different animals. Civil War armies were almost exclusively composed of line infantry, light cavalry, and artillery. Infantry fought in the line, cavalry scouted and sometimes skirmished, artillery performed most like it's European counterparts albeit in slightly smaller numbers.
European armies of the period, on the other hand, were surprisingly specialized. Line infantry were supplemented by light infantry for skirmishing and sharpshooting, elite infantry that could hold the most important section of a line or exploit a weaknesses in an enemy's line. Cavalry had myriad extra divisions: hussars and uhlans were trained to destroy an enemy once it had been routed, curassiers and uhlans could be used as shock troops to rout said troops, even infantry.
European armies also had professional generals and staffs, not the up-jumped lieutenants and captains that commanded in the American Civil War. The Prussians were even well on their way to a modern battlefield medical system that allowed them to cut casualties drastically and return soldiers that would have died in a Civil War field hospital chop shop to return to the next battle or at least be fit for secondary duty.
Both the contemporary Austro-Prussian and Franco-Prussian war were over in months and European wars were often decided by a single decisive battle. It's often been theorized that the American Civil War was so bloody and so long due to the indecisive outcomes of battles specifically due to the lack of these extra troops. Yes one side won, but the other side retreated in relative order and survived to keep on fighting another day.
On the naval side of things, Britain and France actually preceded the US in building ironclad warships. By the end of the American Civil War any European navy worth their salt had a fleet of them, and not the teeny, tiny coastal and river monitor style ships, we're taking full steam cruising frigates with an entire bank of guns. Britain certainly had their fair share of ironclads (20+ by 1865) and a regular navy that dwarfed anything the US could even dream of.
From what I've read on this subreddit, the American armies in the civil war were not regarded well at all by their European counterparts. From Vonadler's post in one of the linked threads:
It is said that Helmuth von Moltke the elder called the American Civil War "two armed mobs chasing each other through the country, from which nothing can be learned". While he probably never said so (I have never found a reliable source saying it), the quote still captures a lot of the sentiment of how professional Europoean officers viewed much of the American Civil War.
And here's one on the ironclads.
Edit: This is actually a much better discussion for your specific question. Previous comment deleted.
Bottom line from that thread: The Union had more ironclads, but most were only suited for coastal defence. The British had many more ocean-going ironclads. Fighting the Union on Union territory would have been tricky. Fighting on the oceans would have been a foregone conclusion in favour of the Brits.
There's plenty more to be found if you search for the American civil war.