Six Frigates is an excellent book about the origins of the U.S. Navy. It also gives a little insight into how quickly Britain and the U.S. aligned themselves, helped in part by the French Revolution.
The way I understand it, it's not quite so simple.
The US is really into individual liberty and self-governance. There was a sizable monarchist contingent here after the Revolutionary War, and they were involved in creating the structure and adoption of the US Constitution (and before that, the Articles of Confederation).
It wasn't at all a foregone conclusion that the US would end up as a republic.