No, the US does not have a database of citizens or a national ID. There's been long-standing resistance to such a thing, as part of the general American distrust of centralized government and what a tyrannical government could do with that information. That resistance has even been bipartisan, coming more from the left or right depending on who is in power. So there's no quick database check.
However, in practice, if you're a citizen, it's pretty easy to prove it. You'll be carrying forms of state-issued ID and you'll have a paper trail that makes it obvious. There's rarely much ambiguity over whether a particular person is here legally or illegally, as much as some might like to pretend there is.