First, no I don’t think the U.S. is historically a nation-state. It is a territorial state, if anything. It may be becoming a nation-state by creating a new culture on its territory. Interesting question, actually.
Using language as a sole basis of nationality is also a tricky point, as has been established. But it has often been used as a piece of the puzzle of ethnicity. But it is actually very difficult to find a state founded on language alone, and it is easy to find many that weren’t. Certainly, the State of Israel was not founded on lingusitic grounds, and indeed that was one of the problems at first, since the immigrating Jews spoke all sorts of languages: Yiddish, Ladino, German, Russian, Polish, English…
As for Germany, the sense of national identity has always had a lot to do with language, despite the many German dialects.
In other words, each case would seem to have its own character, and it’s very difficult to compare them all.
I would have to conclude that a state has a right to exist when a majority of its inhabitants today want it to exist. Of course, this can lead to problems with minorities, etc.
But in the end there is probably no perfect answer.