I will begin by observing that the answer to this question is a logical and philosophical one, hence it does not depend on a particular religion or denomination. It is human nature itself that demands utmost respect for all human beings, and hence all people—regardless of religion—are obligated to work for the end to direct abortion in all cases.
I will, however, answer from the perspective of the Catholic Church.
For the purposes of this discussion, direct abortion is defined as any action “tending directly to destroy human life in the womb ‘whether such destruction is intended as an end or only as a means to an end;’” see Evangelium Vitae 62, which is quoting an address made by Pope Pius XII on November 12, 1944, to the San Luca Biomedical Association.
Regarding direct abortion itself
The Catholic Church takes the position that direct abortion can never be justified for any reason. In other words, although there may be situations in which a medical procedure must be made that results in the death of the child (e.g., in the treatment of ectopic pregnancies, or in the case of particularly aggressive cancers), the death of the child can never be directly intended—either for its own sake, or for the sake of some other good. As Pope John Paul II said in Evangelium Vitae,
I declare that direct abortion, that is, abortion willed as an end or as a means, always constitutes a grave moral disorder, since it is the deliberate killing of an innocent human being (No. 62).
Regarding the case of pregnancy that results from rape
Hence, a woman who has been raped (and whoever finds herself in that situation deserves the utmost care and support!), and is pregnant as a result, is indeed obligated to bring the child to term. This obligation arises from the fact that the child in question is a human being—in fact, a human person—with all the rights and obligations that accrue to a human being, simply for being human. The most fundamental right of all is the right to life.
It should also be observed that although rape itself is an unspeakable crime, pregnancy as such is not, nor does it cause harm to the mother. Rather, it is a normal part of human development. Clearly, the way in which this particular kind of pregnancy comes about is extremely traumatic and violent, but the pregnancy itself is not a violence against the woman or her body.
It is true that pregnancy brings with it certain risks, but modern medical care mitigates the vast majority of them. It is also true that pregnancy—especially one that is unexpected and emotionally charged, as in this this case—can be burdensome for the mother. However, the proper response is for society to help her bear the burden: doing violence to the completely innocent child in her womb solves nothing and heals nothing.
(I do not have access to scientific studies on cases of pregnancies resulting from rape—I am not sure that there are enough cases to be statistically significant—however, ample anecdotal evidence shows that bringing the child to term is often a source of healing for the mother. See for example, this story about a mother who was raped when she was 12; a story about people conceived in rape; a story from a woman who at first wanted her baby dead. Moreover, abortion itself is always extremely harmful to the woman who has it done—at a minimum, for the moral damage it causes, but at least in a significant number of cases, it also produces intense anguish and suffering.)
A look at the “violinist” analogy
Let us now look at the analogy presented by the O.P.
The first thing to observe is that it is an entirely canned and unrealistic situation. In reality, there is no medical situation is like this. (In the case of kidney failure, the proper procedure is to give the violinist kidney dialysis.)
Let us, however, take the “bait” and see if the analogy between the situations really holds. In fact, there are important differences:
The violinist is not healthy. He is, in fact, in a very precarious medical situation, and any treatment that will be performed on him has only a limited chance of success. On the contrary, in the vast majority of cases, the baby conceived in rape is perfectly healthy, and so is the mother (at least for the most part—she is generally not at death’s door, like the violinist). Unless someone does violence to him, the baby is fully expected to survive and flourish.
The violinist depends on the kidnapped victim only tenuously for his survival. Once his disease is cured, he can go on his way. On the other hand, a baby (at least before viability) depends radically on his mother for his very existence. In other words, the violinist depends on the kidnapped victim only because (1) he has an illness and (2) the kidnapped person has been connected to him—in other words, he depends only accidentally on the kidnapped person. The baby, on the other hand, depends on his mother (as it is said) essentially—that is just how the baby is made right now, even though he is perfectly healthy.
Comparing not the crimes committed, but only the situation of the victims (person kidnapped and mother) after the crime is committed, the person connected to the violinist is subjected to a much worse burden. The kidnapped person must remain immobile for nine months in a hospital; pregnancy (without diminishing the real burdens that it entails) does not impose a comparable burden on the mother.
Removing the connection to the violinist does not intend to procure the death of the violinist. It intends to remove the victim from a burdensome (and, in this case, unjust) situation. On the other hand, direct abortion does violence to the baby—it necessarily intends the baby’s death.
With these differences in mind, let us look at each situation:
Would a person who finds himself connected to a sick violinist be obliged to remain there until the violinist recovers?
We should observe that—as in the case of abortion—the fact that the person aiding the violinist was placed in that position by violence is actually irrelevant to the question. The question is: now that I find myself connected to a sick violinist (however I got there), may I disconnect myself?
The answer is, “It depends on the circumstances.” (Assuming I am this victim), I cannot summarily pull out the connections, without at least appraising the situation. I have to consider the life of this sick person, and whether my actions would directly procure his death or not.
In this case, however, because his dependence on me is circumstantial or (to use the classical terminology) “accidental,” I would most likely not be obligated to continue the “treatment.” We are dealing with a very sick patient whose life might (or might not) be prolonged by my being connected with him. The use of my kidneys is, at the end of the day, a risky experimental procedure that has a certain probability of success, but is not 100% certain, and might even cause me harm.
Moreover, remaining connected to the violinist is a tremendous burden (one not comparable to pregnancy, as mentioned above). It could only be imposed on me with my consent—which is obviously lacking in this case.
In summary, the dependence of the violinist on the kidnapping victim for his survival is too tenuous or “accidental” for it to impose an absolute obligation on the kidnapped person to continue the “treatment.” However, even in a case like this the life of the violinist must be preserved as best as the situation allows.
May a woman who finds herself pregnant as a result of a rape do violence to her child?
No, never. There is no way to “disconnect” the baby without necessarily bringing about his death. The dependence of the baby on his mother is essential—brought about by his very constitution.
Moreover, the burden imposed by the baby on the mother is one that can be reasonably be expected to be born. It does not threaten her life or her wellbeing, and what risks there are can (in the vast majority of cases) be mitigated. The evidence strongly suggests that pregnancy and birth are healing experiences for mothers that are victims of rape—and that aborting such children, on the contrary, causes the mother anguish and unspeakable suffering.
A better analogy
I will propose a better analogy that is closer to the situation of pregnancy that results from rape.
Suppose that through some sort of act of violence (perhaps a terrorist attack or something similar), I encounter a newborn baby who has no one to care for him. (Suppose that he is the only survivor of a terrorist attack.)
Supposing that there is no one else who can do so, would I be obliged to take care of him? I think that everyone would agree that the answer is “yes.” At a minimum, I would be obliged to find someone else who would be willing to take over that responsibility.
Does the fact that the situation was imposed on me by violence take away my responsibility? Not a bit. It may make it more difficult for me to take on that responsibility, but the responsibility remains. The proper response, however, is to seek help and support in fulfilling that responsibility—not to do violence to the innocent child who is the subject of it.