I've heard various cases made that the US civil war was "the first modern war" for various reasons, including that the entire country was mobilized to fight, it wasn't just army against army, but nation against nation, also the strategy of "total war" in Sherman's march to the sea, destroying the confederacies ability to manufacture, and Grant's strategy of defeating the south through attrition rather than military victories.
Is there any validity to this claim, or is it just Amero-centrism?
I'm not asking if the war happened in this way, but rather was it influential in the course of warfare afterwards? Did other generals and countries see these events and use some of the strategies first used the the US civil war?
There are two aspects of this. Did the US Civil War have aspects that could have been studied for the future? Certainly.
Did other modern nations think so at the time? No.
There's no evidence that the quote about the US Civil War attributed to Helmuth von Moltke the elder (the Prussian man behind the victories against Denmark 1864, Austria 1866 and France 1870) was ever uttered by the man (at least I have not been able to find any credible such), but it does capture some of the spirit of what the European militaries thought of the US Civil War: "Two armed mobs chasing each other around the countryside, from which nothing could be learned."
The US Civil War was a war fought by amateurs, led by amateurs and conducted in a most unplanned, amateurish way according to the Europeans. Why?
First of all, there were only 534 West Point graduates that fought in the US Civil War - and while there were also militia part time officers on both sides, both sides had a distinct lack of officers to command the roughly 1 500 000 men that were in the field at any given time. Men who raised volunteer units often commanded them themselves and appointed or let the men elect their officers and NCOs, usually men without any previous military experience. Most of the generals in the US Civil War had no experience in staff work, meaning they had not spent time with mobilisation plans, supply and movement tables by railroads or organising logistics. Men such as Halleck, Grant and Sherman had left the military at the rank of Captain to pursue civilian careers before the war. Lee was one of the few US Civil War generals with experience in logistics, having served as quartermaster general of Scott's force in Mexico, organising the supply for a force that never exceeded 8 000 men.
European militaries had a professional core, with officers that had gone through both an officer school and cadet training and general staff training. They had experience working with mobilisation plans, railroad tables, logistics and modern tactical theories.
While the Europeans had shifted to rifle chain or rifle swarm tactics, with the platoon as the manouvre unit, the US Civil War armies used Napoleonic tactics with the battalion as the manouvre unit on the field.
The Europeans also fielded a strong cavalry arm capable of pursuit - for example, roughly half the 20 000 Austrian prisoners taken at Königgrätz 1866 were captured by the Prussian cavalry in an hour after the Austrian army started to retreat. The combination of a strong cavalry arm to conduct battlefield pursuit, prepared and precalculated mobilisation, logistics and marching tables and always keeping a large force of professional troops in reserve meant that European battles tended to be far more decisive than American ones in this era. The victor in an American battle was usually so sapped that no proper battlefield pursuit was possible while the losing European army was mercilessly pursued until it was no longer a fghting force, or holed up in a fortified postion from which it could be staved out.
The Europeans also had a strong, professional artillery with strong traditions of standardisation. The French were especially keen here, and used rifled guns of the La Hitte system against the Austrians in Italy 1859, causing the Austrians to switch to an all-rifled artillery that served them quite well at Königgrätz 1866.
The Prussians introduced a telegraph corps in their army in 1830. The French moved around 100 000 men by rail into Italy before the Battle of Solferino 1859- Trench warfare had been common in European sieges since the early 1600s. The French used ironclads of the Aetna class against Russian fortifications in the Crimean War. The French and British used muzzle-loaded rifles with minie balls in the Crimean War and the Prussian used breech-loaded rifles in their wars against Denmark, Austria and France.
The only development in the US Civil War that actually attracted a lot of attention from the Europeans was the first combat between two ironclads at Hampton Roads 1862 between CSS Virginia (built on the hull of the USS Merrimac) and the USS Monitor. The Europeans would experience the first fleet combat between armoured ships when the Italians were defeated by the Austrians at Lissa 1866.
What really could be learned from the US Civil War was how to quickly and effectively create armies out of nearly nothing, and then the arderous process of increasing professionalism, structuring logistics, weeding out bad officers and generals, introducing effective drill and mobilise society to support it, from nearly nothing.
So, to answer your question, no the war was not influential, as Europeans thought they had little to learn, since their battles and wars were far more decisive.