Can they? Yes. Do they? Not often.. A high school diploma wont get you very far, so its rarely asked. Most jobs that require a higher education ask for your college transcripts sent directly to them, from the college..
Sounds like your asking if they can somehow prove that you did in fact graduate HS (or not) or obtain your GED. Unfortunately I don't have the answer for you. I've always wondered how do they KNOW for sure if you've graduated or just said you did.