After doing some research on these forums and elsewhere...
I haven't found anything that specifically says hospitals can check your FBI records, but it seems like any company that provides care to children, the elderly, or the disabled is allowed to do so. Nursing homes and home health care agencies are definitely allowed to fingerprint you and get your FBI rap sheet, and at least some states REQUIRE them to do so. Same thing for schools.
Depending on your record, you might not be legally prevented from working, but that doesn't mean anyone is going to give you a job.
It's weird that after spending a lot of time looking into a career in health care, I never came across anyone saying they had to be fingerprinted to get their job.
One of the problems with the FBI rap sheet is that it may only show your arrest with no record of outcome. If your charges were not felonies and happened a long, long time ago, it's possible that the state and your county courthouse might not have your records anymore. If your charges were dismissed or you need to document the outcome for any other reason, I don't know what you're supposed to do.
Anyway, if I were able to document everything, I'd like to think I could get licensed, but I don't think there's any way I could ever get a job at a hospital. Maybe no job anywhere.
I think I've only found one person here who admitted a criminal past and got hired anywhere (and that person soon quit because the facility was breaking all sorts of regulations). If anyone knows a person that actually got hired with a criminal past, I'd like to hear about it.