iwanna, I think you did the right thing, by acting immediately on this.
But IMO, you should call the hospital TODAY, just to have peace of mind they will not act. There business office is open today, I'll bet.
At least they can make a notation on your records, that you've paid, with the ref.#. Do a follow up call on monday, to verify they've received it.
That's what I'd do, but that's just me. I trust no one, no thing, only myself, when it comes to stuff like that. I have to satisfy myself fully, before I can relax. But again, that's just me, and what I'd do.