This apparently is not an easy case as one might suppose.
According to this site, here is the reasoning of those who performed the abortion ( it seems very reluctantly ) :
EXCERPT:
http://www.catholicnews.com/data/stories/cns/1005213.htm
Linda Hunt, president of St. Joseph’s, said in a statement after the bishop’s news conference that the hospital was “deeply disappointed” by the action but would “continue through our words and deeds to carry out the healing ministry of Jesus.”
In May, officials at St. Joseph’s publicly acknowledged that an abortion occurred at the hospital in late 2009. The Arizona Republic, in its initial story on the matter, also revealed that Mercy Sister Margaret McBride had incurred an automatic excommunication because of her role on the ethics committee that sanctioned the abortion.
“Consistent with our values of dignity and justice, if we are presented with a situation in which a pregnancy threatens a woman’s life, our first priority is to save both patients,” Hunt said in her statement. “If that is not possible, we will always save the life we can save, and that is what we did in this case.
“We continue to stand by the decision, which was made in collaboration with the patient, her family, her caregivers and our ethics committee,” she added. “Morally, ethically and legally we simply cannot stand by and let someone die whose life we might be able to save.”
The bishop's job is to stand for what the Catholic Church believes and teaches, and there it's really quite simple:
Under no circumstances can an innocent* person be deliberately killed, not even to save another person's life. No exceptions.
*A conscious aggressor is not an innocent person, thus self defense is allowed. An unborn baby is an innocent person, however.
The hospital can go along with Catholic teaching or not, but if their choice is "not," then it's dishonest for them to keep calling themselves "a Catholic hospital".