If you are in the hospital on shabbos, of course the best thing to do is to stay until after shabbos. While a Hatzalah member is permitted to return because otherwise we are afraid he won’t go to begin with, as 80 said, I think this only applies to the Hatzalah member or a doctor, but not to the patient himself.
As far as going to the hospital on shabbos, if it is a serious emergency Hatzolah should be called as they are the quickest responders, but if it’s not a life and death thing but you need to go, I think local EMS or a cab may be called. Not sure how to pay a cab though.