Okay, so I somehow did such on another file, but I don't quite recall how. In a game file on my other computer, I somehow defeated the French, or at least got them to withdraw from America/The New World. Can anyone go into detail on how exactly that's accomplished?

You cannot defeat them, sadly. No matter how many colonies you conquer, they'll continue to pop up every now and then and create new cities. But there is a random event that can happen where one of the foreign nations will withdraw from the new world and cede its colonies to another nation, typically the result of a peace treaty between two nations after a long war.

Gnarbrag is correct. It is not random but a fired event that occurs at 50% rebel sentiment. The smallest CPU power (as defined by pop) is absorbed by the largest.