While the US army 1865 was powerful, and certainly the best force in North America, and probably capable of beating any other army on US territory, I would not go as far as to say that it was the most powerful army in the world.

"Total War" as you say it had been used since medieval times in Europe, with both the 30 years' war and the Napoleonic War being more recent examples of aiming at destroying the enemy's ability to fight by removing his supplies and the means of production of such supplies from his territory.

Ironclad ships were used in the Crimean War, and both France and Britain had comissioned sea-going ironclads before the US Civil War started. In fact, due to the speed, armament, armour and tonnage of the sea-going French and British ironclads in 1865, one can safely say that a single one of them could take on the entire US and CS navies (all ships ever used during the Civil War).

The French used massive railway transportation to move their army into Italy to fight the Austrians at Solferino 1859. Railroad transportation was standard staff work in European armies since the 1840s.

The Prussian army had a telegraph corps since 1837.

I'd say that all thigns equal (fighting on neutral territory), the US army comes up short against the French, the British, the Prussians and possibly also the Austrians in 1865 because:

1. It lacked a dedicated staff to plan mobilization, movement of troops, logistics and strategy, which all European countries had at the time.

2. Its armament was heterogenous, with a vast array of different rifled muskets (Lorenz, Enfield, Springfield and many others) and an equally vast amount of different artillery. Much of the artillery was still smoothbores, and the rifled artillery had substantially shorter range than its European counterparts. The 3" ordnance rifle had a range of 1 673 meters, while the British 12pdr Armstrong rifle had a range of 3 100 meters, the Prussian C/61 and C/64 rifled artillery had a range of about 3 700 meters while the Austrian rifled artillery had a range of about 3 750 meters while the French guns of the La Hitte system had a range of about 3 000 meters.

3. The US lacked the industrial know-how to cast iron and steel with no hidden imperfections, which meant that rifle barrels and artillery pieces had to be produced in wrought iron, which was weaker and much slower to make (as an example, the Prussians procuded 270 000 Dreyse rifles before 1862, when they introduced a cast steel barrel, and 900 000 between 1862 and 1870) and continued to rely on deliveries of European weapons and supplies to equip its men.

4. The US cavalry was incapable both of screening their own army's movement and conducting a proper battlefield pursuit. At Königgrätz 1866, the Prussian cavalry took 20 000 Austrian prisoners in two hours after the Austrian army started to retreat, only being stopped by a counter-charge by the Austrian cavalry.

5. The US tactics were still centered around Napoleonic types. The battalion, a massed formation of men, was the main manouvre unit and the troops fired salvo fire at short ranges. The Europeans either used the rifle chain tactic, with the platoon as the manoubre and fire unit, or the Prussiabn rifle swarm tactic, in which men moved and fired indvidually, reloading and firing from cover, sitting or lying down as the Prussian Dreyse rifle was breechloaded.

Considering these factors, if a US and a European army meets in neutral territory 1865, something along these lines will happen:

1. The Europeans, having a much more powerful cavalry force, will have much better recoinnasance before the battle.

2. The Europeans, having dedicated staff dealing with such matters, having trained it extensively, will be able to use roads and railroads to concentrate their forces quicker.

3. The Europeans, again having a much more powerful cavalry force, will be able to screen their movements much better and may catch the US force unawares.

4. The Europeans, having artillery with twice the range of the US artillery, will be able to dominate the battlefield with artillery, destroying US artillery as it tries to move up and disrupt US infantry formations as they try to draw near.

5. The Europeans, using platoons as manouvre and fire units will be able to disrupt the US battalions as they aproach.

6. If the Europeans are the Prussians, they also have a much, much higher rate of fire and will fire individually and from cover, outfiring the US infantry by a ratio of about 4-1, and doing it from cover at men standing in line.

7. Once the European force has won, its cavalry force, being much stronger than its US counterpart, will be able to conduct a proper battlefield pursuit, utterly shattering the US force. If the US force would win, despire the earlier points, the European cavalry can screen their own forces retreat to allow them to reform for the next battle.

The French already fielded a 700.000 men large army 60-70 years before that under Napoleon. People also often forget about the Crimean war when discussing 19th century warfare where the Russians were forced to muster nearly 900.000 men. The Franco Prussian war started in 1870 and the Prussians had a 1.5 million strong military fighting againdt 2 million French men (total deployments).

The 19th century is often seen as peacefull by people who have only basic knowledge of the period. There were some very bloody European wars, the American civil war and many colonial wars. To answer your question: no. The Americans didnt have the most powerfull army in the world. You could compare the US civil war with the Spanish civil war in the sense that it was indeed a testing ground for new weaponry and doctrines. European advisors would travel to the US and witness the horrors of the US civil war first hand and take the knowledge back to Europe.

The US had a total population of around 30.000.000 in 1860. To give you an idea the Austrian Hungarian population consisted of around 37.000.000 20 years before that already.

Besides Europeans powers that ive already mentioned like the Russians, Austrians, French and Prussians, there is still the elephant in the room. 1865 is still within the golden years of the British empire. Even if the US had the most powerfull army in the world in 1865, they couldnt do anything with it to threaten other continents, as no navy stood a chance against the British. The British Empire is still the worlds greatest power in 1865 with even Canada still firmly held by London. (Canada only had a population of around 3-4.000.000 around this time).

They were indeed strong, but American didn't really come into its own until at least WW1, in which the other powers were so weakened, America had few competitors, though France, Britain, Japan and the Future Soviet Union remained strong.

The most obvious one to compete with, is Prussia. This German state had, since the time of Peter the Great and further back, been lending out its armies for usage by other German states of the Holy Roman Empire, and had gained valuable experience and power. Prussia would come into its zenith, in 1866, after the Austro-Prussian and Franco-Prussian Wars signified that Prussia's time had risen, and they became central to the power-base of the newly unified German Empire. Prussia forces then were to become a major part of German military operations. Their defeat of the French army marked their time as the top land power within Europe, though it was achieved with assistance by the help of the smaller German states.

The British Empire also comes to mind. While emphasis was placed upon their massive, all-powerful juggernaut of a navy, the British army had much to draw upon with land forces. Of course there were the standard Red Coats, but there were also the Scottish Regiments, such as the thin red line, that held off numerous assaults thanks to their formation, despite being out-numbered overwhelmingly. The British had forces to draw from Australia, Canada, New Zealand, Ireland, and the Sepoys from India, and Gurkhas of Nepal, the latter being especially fearsome warriors. All of this would allow the British Empire to maintain control over its colonies and defend its interests in war.

The other powers, however, couldn't really measure up. After the Crimean War, the Russians were left greatly weakened, even the Son of the Czar and Heir to the throne admitting they had become a second rate power. France during Napoleon the Third had lost its military strength, much of its army during the Franco-Prussian War made up of Conscripts and criminals forced into the ranks, explaining their poor performance and morale during the battle. The Ottoman Empire was in sharp decline, in massive debt, and unable to fund its education. Its military, was not even allowed to conduct war games and practice at the time.

Just in terms of the United States alone, I would question how powerful the US army was in 1865.

In May 1865, the Union Army at its Grand Review in Washington, DC was arguably at its height, having a national parade as its "benediction" as a victorious force of some one million.

But even the Grand Review itself was a first stage in demobilization. One needs to remember that the professional US army as of 1861 was around some 15,000 troops - everyone else in the Union Army was a one-year or three-year volunteer, or part of a regiment raised for a three-year term by state. These state regiments were roughly analogous to (but not the same as) modern National Guard units, in the sense that they were state-based groups that were "federalized" for a set duration - the members of the professional army were not mixed with these groups. When demobilization came, those that were serving longest were the first to be discharged from service. Lincoln had initially called for 75,000 men to volunteer in 90 day regiments in early 1861, but it quickly became clear that more troops with longer service periods would be needed, so Lincoln and eventually Congress authorized calling up about a million volunteers for three-year service periods (confusingly, some states also offered two-year service periods that the War Department reluctantly agreed to). State governors and legislatures even authorized, raised and equipped some units at state expense, although this was only at the very beginning of the war (the Quartermaster Bureau in the War Department ultimately took this over). The Union also instituted draft call-ups in 1863, but this was in short something of a mess, and ultimately only 46,000 of those drafted served (another 74,000 provided substitutes).

The US government ultimately could not pay to maintain such a large force beyond the duration of "major combat operations" (I am intentionally using an Iraq War term here). Something like two thirds of Union expenses during the war were funded by debt, and the government was very anxious to both cut costs and to begin paying off this debt. As a result, demobilization was a very rapid affair: the Union Army dramatically decreased in size from about 1 million in April 1865, to 125,000 in November, to 90,000 at the end of January 1866. Between May 1 and November 15 some 801,000 officers and men were discharged, with 641,000 being mustered out between May and August alone. Those forces that remained had very restricted mobility (often relying on rail links or just plain travel by foot) as the government also tried to decrease expenses with a mass sell off of war horses.

Of these remaining troops, some 50,000 or so were under the command of General Sheridan, and spread along the Southern border - or to "pacify" native peoples on the Southern plains (the Battle of Washita against the Southern Cheyenne is an infamous example of this). These troops on frontier duty tended to be disproportionately people of color (for example the "Buffalo Soldiers" of the 10th Cavalry Regiment) or very recent European immigrants, and as such were considered to be the lowest of the low in American society (even by their own officers). This is in contrast to what James McPherson writes about the Union army during the civil war, where contrary to popular perceptions Union forces were *not* disproportionately poor, immigrant, or conscripted, but disproportionately white-collar, native Protestant, and volunteers of 1861-1862.

This left painfully few troops available for the other major duty it had post 1865: maintaining a semblance of peace and order as an occupation force in the postwar-South. The area of the former Confederacy was the size of Western Europe (some 750,000 square miles), with a population of nine million, and with the US army spread out in 324 garrisons and 630 outposts. Troops were stretched dreadfully thin - some 100 cavalrymen were available for duty in all of Mississippi. This meant that the US army was woefully unprepared politically, logistically, militarily and morally for maintaining peace and enforcing Reconstruction laws against "night riders" or, again to intentionally borrow Iraq War language, networks of terrorists and insurgents. Garrisons dwindled ever-further (sometimes numbering in the tens or even *ones*), and were completely unable to contain the racial violence and guerrilla warfare in the South that "Redeemers" (post-Confederate white Southern groups) waged against US military forces, pro-Reconstruction whites, and blacks alike. Most of the defense offered against these attacks were attempts at local self-defense, or through Reconstruction Republican state governments. By 1877, the very last US troops were withdrawn from the South (from South Carolina and Louisiana), which more or less prompted the collapse of the remaining Reconstruction Republican governments there. No US troops would return to enforce federal civil rights laws until the 1950s.

In summary: the US army appeared mighty in 1865, but it was largely a paper tiger - funded by US government debt, staffed by volunteers whose terms of service were almost up and were to fight a very specific type of conflict against the Confederacy. Even by late 1865, the size of the army was being massively reduced, with even its means for mobility drastically curtailed. The US army in 1865 and after was, depending on the situation, unable to enforce peace and order in the areas it occupied (the South) or with only great difficulty and with great civilian casualties and suffering able to enforce its order (the West).

Sources:

James McPherson. Battle Cry of Freedom

Richard White. The Republic for Which It Stands: The United States during Reconstruction and the Gilded Age, 1865-1896

Eric Foner. Reconstruction: America's Unfinished Revolution, 1863-1877

William Holberton. "Demobilization of the Union Army: 1865-1866". A Lehigh University master's thesis that I found and that was a very interesting read, can be accessed here.

ETA: For some broader perspective - even in the immediate aftermath of World War II, the US military was overwhelmingly demobilized. The idea of the US maintaining a large, standing army in peacetime didn't come into being until after the Korean War, and the idea that these forces would be professional volunteers didn't come about until after the Vietnam War.

Allthough population is a very limited measure of military power in any period, it is especially so before WWI because the complete mobilization of the population or in clauzewitz terminology, 'total war', had not yet been entirely realised. During this period, superiour logistics, tactics and organisation was still the 'name of the game' on advanced battlefields. Just as Napoleon vanquished all of Europe with his bold maneuvers and incredible speed at the beginning of the century, Helmut Von Moltke the Elder proceeded to crush all opposition in the mid-century. Being among the first to effectively utilise trains for large scale troop transport (just like Napoleon used his roads), Von Moltke led Bismarcks armies to crush France, Austria and Denmark in rapid sucession. Thus carving modern Germany out of a collection of smaller feudal states. This was an enormous feat. Austria presented a considerable force believed to be far greater than that of Prussia at the outset of war, while France was generally still regarded as the largest and best military force of the era. Despite Napoleons eventual defeat, It's military traditions and size had largely been preserved. Denmark too was believed able to put up a fight. Yet, the Prussians smashed all opposition with extreme efficiency.

Indeed, the best measure of actual military capacity is war between the parties in question. Here we have Prussia beating two of the top four landpowers at the time (that is, Russia and Britain). Allthough there was of course no war between Prussia and the United States to test their capacity under ideal consitions, I think this sufficient to make it a fair assessment. Indeed, I would say that as the U.S. did not feature among the five greatest powers in terms of resources at the time and had both limited military traditions and manpower, the Prussian Army was likely much stronger. Probably also the other great powers. That is, if you actually placed them on the same landmass.

Indeed, during Bismarck and Von Moltkes reign, Prussia was pretty much unbeatable.