In my American history class we were taught that at one time, France and Spain had a presence in the Americas and France had even planned to try to take over the colonies before they had a revolution of their own. This got me wondering to how the established powers such as Spain, France, and Britain viewed this colony that overthrew their imperial control, quickly grew and amassed power, had civil war, and had the audacity to pick a war with a power (although much reduced by then) in Spain in these different phases of America's history. I'm especially interested in a decade or so after Revolution had ended and France and Spain had territories in the Americas.
America did not overthrow its imperial control on its own. It did it explicitly with Spanish and French help. France in fact declared war on Britain during the Revolutionary war and was allied with the United States during the war, this is evidence by the fact that the treaty which ended the revolutionary war was signed in Paris. Spain was also allied against Britain during the Revolutionary war and supported the USA. The Dutch was also conspiring against the British on behalf of the colonies. European aid was vital to America gaining independence, without it, it is likely America would not have had nearly the success it did.
I'm confused by your question but if I'm correct you are assuming that the USA overthrew Britain by itself and not with the help of European nations and what where those European nations reactions to the US gaining independence? Those nations are the same nations that were the reason America gained its independence in the first place, surely you were taught this in your class also?