There were several attempts at taking advantage of the US being distracted during the civil war, but as far as I know, none after the civil war. One needs to remember that before the civil war, the US navy was weak, ranking in somewhere 7th to 12th in the world, depending a bit on when you compare - Britain, France, Russia, the Netherlands, Spain, the Two Sicilies all had larger and stronger navies, and at times Sweden-Norway, the Ottoman Empire, Portugal and even Austria could compete with the US navy in power.

The US army was by European standards pathetic before the civil war - the regular army was around 16 000 men just before the outbreak of hostilities - as a comparison, the Swedish (a minor power at the time) army was some 26 000 men, while the grand powers had armies in the multiple hundreds of thousands in this era.

At the tail end of the US civil war, the US fielded one of the largest navies of the world (although it was almost exlusively smaller ships for raiding and blockade enforcements, with a single sea-going ironclad and six stram frigates for heavier vessels) and a huge army of around 1 million men. The US had also learned quickly from the experience of the war and had formed an impressive supply and logistics organisation, expanded her military industry to meet the needs of this huge force and in general raised the level of professionalism and discipline among officers and men alike to grand power standards. The US still lacked a strong cavalry establishment and an extensive general staff and their associated planning, but in general, the US was militarily more powerful than she had ever been and I dare say that she would not reach the same level of military power - absolutely and relative to other powers - until 1918 after mobilising for the first world war. While the US would quickly disarm after the war, she would not go back to full peace time status until the end of the reconstruction in 1877.

That said, in general the European nations respected the Monroe doctrine before and after the US civil war, especially as this is the era of Pax Britannica - doing something on the international stage that was contrary to British interests were generally a bad idea. And Britain liked the US as a wealthy and stable trading partner, as cheap US grain and cotton fed British labourers and the cloth industry they worked in, earning both parties large profits.

With the US distracted and unable to uphold the Monroe doctrine, Spain attempted to force Chile and Peru to comply with its demands in the Chincha Islands War 1864-66 and also re-colonised/occupied Santo Domingo/the Dominican Republic 1861-65.

Likewise, multiple European powers, including France, Austria, Belgium, Britain and Spain intervened in Mexico - although Britain and Spain withdrew as they learned the French intended to occupy all of Mexico and introduce Austrian Archduke Maximilian as Emperor of Mexico. The French intervention lasted from 1862 to 1867, after which US-supported liberal rebels under Benito Juarez ousted Maximilian's regime and his European allies.

In general, the US was an economical powerhouse with vast potential for military mobilisation, but with the friendly relations and profitable trade with Britain and no powerful enemies in America and no European powers beyond Britain and perhaps France capable of projecting power over the Atlantic, the US did not need a strong military establishment and were compared to contemporary grand powers weak, especially when it came to the army.

As you can see, the US was at its strongest immediately after the civil war, but it was during the war, with the US distracted, that various European powers tried to take advantage of it.