Yes, many hospitals began with a church. My sister works at Good Samaritan Hospital, where they still have nuns walking around in their habits. I have worked at two hospitals, Good Shepherd and Sacred Hearth, both Catholic but without the nuns. Some have more or less kept to the religious ethics. Sacred Heart in particular was very strict about the care they would provide, both in the hospital and for employees through the medical insurance. For example, they would not provide abortion even in to save the mother's life, one could not even mention birth control, and they would not allow withdrawal of medical treatment in terminal patients.

Now that I am trying to think of more examples, I can only remember Catholic hospitals, not other religions. Curious.

Oh, yes, in France it's 'infirmier' and 'infirmière'. Wonder what it's in Quebec...