I have heard the idea thrown around from some people that the American military pushed innovation more than Europe in the 19th century... They referenced things like the Gatling Gun, less heavy reliance on cavalry, and development of ironclads.
How accurate is this? If it is inaccurate, what sort of military innovations did Europe develop in that time?
In the case of small arms, definitely not. The Federal army in the Civil War had used mostly muzzle-loading weapons, like the 1855 Springfield rifle, but had some units ( notably cavalry) equipped with more modern breechloaders like Sharps, and some of those , like the Spencer, had magazines that enabled the soldier to fire several times rather quickly. When the war was over, trials were conducted of various designs ( like the Sharps and Spencer) that might replace the muzzleloaders. But instead of making a leap to magazine weapons, the War Dept settled for a stop-gap: a breechloading conversion of the muzzloading rifles it already possessed. This was de-bugged and improved over the next few years to become a rifle in its own right,the Trapdoor Springfield.
Meanwhile, breechloaders had been already tried in Europe, with the German Dreyse Needle Gun in the 1840's, and by the 1850's most European countries had some sort of breechloading rifle as standard. But the 1870's would also see the arrival of magazine weapons, like the Swiss Vetterli , which was closely followed by the Germans ( in 1884, with the 71/84 Mauser) . The French even jumped further in 1886 with the first rifle to use smokeless powder. This was immediately imitated by the British ( with the first Lee-Enfield) and the Germans in 1888. By the 1890's, therefore, the magazine-fed smokeless rifle was pretty standard in Europe. By contrast, the US clung to the single-shot black-powder Trapdoor Springfield. When it did adopt the magazine-fed smokeless-powder Krag in 1894, it was already somewhat obsolete, as the newer German Mauser models could be reloaded much more quickly with stripper clips. The US Army discovered this when it came up against Spanish troops in Cuba, during the Spanish American War, who could deliver much higher rates of fire than the Americans could return.