Then your next job calls this job to verify your employment, finds out you quit without notice, and withdraws the offer
Edit: I get that no one cares that what I'm saying is a real thing that happens in some companies. Just know that it is and tread carefully if you're thinking about quitting with no notice period
Edit: Also no, it is in no way illegal in the US for your previous employer to reveal a basic description of the terms of your separation to future/prospective employers