Either the Catholic hospitals are essentially public hospitals with some statues and images of the Virgin in the lobby and corridors, to give a superficial Catholic atmosphere, or they are genuinely Catholic institutions operating in accord with Catholic convictions.
If the former, then they have to provide services any public hospital does.
If the latter, then they should be exempt from having to provide abortions or any other treatment not in accord with Catholic faith – but should receive zero public funding.
I started out this discussion under the mistaken impression that Catholic hospitals were self-funding, or perhaps received a small amount of state funding but retained, by agreement with the state, complete autonomy on all policy matters. It appears that I was mistaken about the facts.
The way I see the matter now is that all general hospitals for general community care should be secularized entirely, removed from all religious control or even religious affiliation (out go the statues of the Virgin, unless Stars of David etc. are also allowed in the lobbies and corridors), and that the place for Church-controlled health-care institutions is covering special niches which the Church sees to be uncovered by the public arrangements. All Americans should have a core of basic rights to medical care, and that should be covered by the secular state, not by churches; but beyond that defined core of medical rights, churches and other religious institutions should be free to offer additional medical services, to which no one has a “right” but which would be greatly appreciated by some, out of charity, out of a desire to see no one fall between the cracks.
There is precedent for this in many areas. The state provides welfare in various forms for the poor, but in addition, many churches provide, e.g., “out in the cold” programs for the homeless in the winter. Analogously, if state care leaves some people without some medical assistance that it would be good for them to have, medical assistance that could be given from church-run, relatively inexpensive small clinics (as opposed to massively expensive large hospitals) would serve a social good. Such auxiliary services would not be required to perform abortions or do anything else contrary to the Catholic faith. I think it would be better for both the Church and the state to adopt arrangements of this sort, than to continue with a system that relies on “ambiguously public” and “ambiguously Catholic” institutions to deliver basic health care.