Translation:He is not a patient of the hospital.
English grammar question: when do you use the extra genetive markers with "of"? You say "a friend of mine", not "a friend of my", "a patient of his" , not "of he" or "of him" ",a daughter of hers" and "a room of one's own" and so on. I suppose you could be an admirer of the hospital, an employee of the hospital or a critic of the hospital without the genetive s. So, the question: why might he be "a patient of the hospital", but not "of the hospital's"?
"A friend of mine," etc uses a pronoun (my, him, her, etc). "A room of one's own" uses the possessive form (that 's). The room belongs to the person.
This particular phrase is a little weird, i think, because it uses "of," where most native speakers (at least in the US) would say "at." I am a patient AT the hospital. I am a guest AT the hotel. Using "of" does indicate that it should use the possessive form, but it's...just not used, typically. It may be because people can possess things, but objects can't (at least in this structure). A hospital can't possess me, but i can possess a room in a hospital.