There was not a rapid rise, it took a while, and the United States was pretty dull for over a hundred years. For the first few decades, the United States didn't have much of a military. The initial impulse to gain the military power to project elsewhere, was to fight pirates in Northern Africa which were impacting merchant shipping. The US, already with a strong merchant marine tradition, built 6 ocean-going naval warships, to protect US shipping in the Mediterranean region.

Civil War era weaponry and military, while large, was largely designed for use exclusively in the North American continent. While combustion powered warships with armored turrets (as opposed to engine augmented sailing ships with broadside weapons) were innovative, and you can see the beginnings of 20th century warships in John Erricson's designs, the ships had low freeboard, and were entirely unsuitable for ocean travel, and not designed for a superpower to project military power.

The first world power stroke may be the Spanish American War, which resulted in a rather obvious message to the rest of the world in the White Fleet travelling around the world in the following decade. However, there were very strong isolationist tendencies in the US for decades after the American revolution, which you also see through the hesitancy to enter WWI and WWII.

There was absolutely pressure from England in the early 19th century, which resulted in another short war, but the American aspect of that conflict didn't spill much outside of North America, so I wouldn't say (military) pressure from England influenced the United States into becoming a superpower capable of projecting its military around the world.

I think we became a world power coming out of WWI and a superpower during WWII.