Relax. It's a bureaucracy.
I once waited more than three weeks for a formal offer after a job hunt after two stellar interviews and being told with a wink "you're in" at the end of the second. We were under contract to purchase our house; if we didn't close on the mortgage soon we were going to lose it.
And this was in a *small* healthcare system.
The bad news is, you're going to have to deal with the suspense for now. One day is nothing. You're on someone else's time, and they already HAVE a job.
The good news is, if it's gotten this far they are serious about you. There may be references to check, licenses to verify, paperwork to file, blah blah. Good luck!