No government entity is financially relying on a Catholic hospital to care for its citizens. If the Catholic hospital closed, it would reopen the next day under new management (quite possibly government management), since most of the money it survives on is government money. The Catholic Church is certainly not pouring money into hospitals. Various Catholic agencies are in the hospital business to MAKE money.
As for NYC schools, the Catholic schools are closing bit by bit, so the slack will have to be taken up by public and other private schools.
So what happens if the hospital closes and the diocese sells the property to someone else?