Do Catholic hospitals have a right not to perform abortions?

Actually, the Catholic Church has always held that abortion -- understood as deliberately ending the life of an ensouled human being in the womb -- is gravely immoral. The changed view to which Andrew refers is the result of changed understanding of when ensoulment occurs. Remember that relative to the history of Christianity, biology has only recently revealed what happens in conception and gestation; Thomas Aquinas, for example, thought that life began at "quickening," or the mother's awareness of fetal movement -- typically after (what we now know is) 15-20 weeks of life and growth! As for your question, I'd refer again to Aquinas: everyone has a moral right to refuse to act against his or her conscience. To deliberately do what your conscience forbids would be morally wrong, always. (The converse doesn't hold, by the way: it is not always morally right to do what your conscience permits. Your conscience could be in error. But on the negative side, even if it is in error, if you do what it...