Catholic hospitals do have the right to refuse to perform abortions and other procedures that go against Catholic teachings (e.g. vasectomies, tubals).
Individual Catholic doctors practicing at other hospitals also can refuse to perform procedures that go against Catholic teachings. My more-Catholic-than-the-Pope parents practiced at a city hospital. There was a list of doctors who would not participate in abortions, and my parents, along with most of the Catholic and Orthodox Jewish doctors, were on it. My parents were also the only two doctors in the entire hospital who refused to participate in tubals. But there were plenty of doctors who had no problem with abortion. TOPs and TLs would get assigned to one OR room and residents and attendings with no objection to abortion would be assigned that room.
About the case described in the OP: My understanding of Catholic teaching is that abortion is permissible if the pregnancy poses an imminent threat to the pregnant woman's life. Even my aforementioned more-Catholic-than-the-Pope parents would not object to participating in termination of an ectopic pregnancy. Catholics consider the fetus to be a separate independent life, starting at conception, but the fetus isn't going to survive anyway, so the question is - abort and have one death, or don't abort and have two deaths. Apparently the Vatican considers the latter preferable.
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