Do Catholic hospitals have a right not to perform abortions?

Well THAT'S a big question, I won't attempt to answer. But I might phrase it differently: 'should' they have that right ... since I prefer to assume that we get to decide which 'rights' to apportion, rather than that we somehow discover the 'rights' that already exist .... And to begin an answer to such a large question you'd have to begin working out very general views on the nature of morality in general, as well on the relationship between public and private, and more -- and no doubt you might like to distinguish between such cases as "purely" elective abortions v. those which are in some sense medically-motivated or "necessary" (eg threatening the life of the mother) ... Even more interestingly, esp with respect to the latter distinctions, you might want to explore exactly how it is that abortion has come to be so vehemently opposed by the Catholic hierarchy -- in fact I recently was reading that until the 20th century the Church's opinion on abortion in general was far more liberal and lenient...