I do not really know, but would suspect the two share the same root.
Since hospitality represents the kind and warm treatment of guests and that is quite what I expect from a hospital.

Being a fan of malapropism, in my younger days, I recall a radio "character", Bill Corsair, (he could be referred as an 'on-air personality' but he had a lot of "character") in Philadelphia who used "Horse-Pistol" when referring to the hospital.

"I am certain there is too much certainty in the world" -Michael Crichton