In the United States, if a health care provider's right to religious freedom is in direct conflict with a law that requires him/her to perform an abortion contrary to their religious beliefs, the courts would apply the legal standard of strict scrutiny to that case. To summarize, the following conditions would need to be met:
- The law must serve a "compelling government interest". In the case of abortion, The Supreme Court ruled in Roe v. Wade that a woman has a right to terminate a pregnancy, so laws protecting that right easily qualify.
- The law must be "narrowly tailored" to achieve that interest. This is where it would get difficult because yes, a woman has a right to an abortion, but the health care provider also has an undeniable right to refuse to perform one because of his/her religious faith. Any law that limits either of these two rights must have some mechanism to accommodate the other.
- The law must provide the least restrictive mechanism for achieving that interest. If there exists any reasonable way protect a health care provider's right to religious freedom, the government must do so to the maximum extent possible that doesn't also infringe on a woman's right to have an abortion. This is where such a law would likely fail, because there's always some other doctor that could do it.
There also needs to be a distinction made between negative rights and positive rights. A negative right is one which you have that cannot be taken away from you. A positive right, on the other hand, is one which requires someone to act on your behalf, or you to act on theirs. Both religious freedom and abortion are negative rights. It's one thing to have a law that protects a woman's right to an abortion. It's another matter entirely to have a law that requires a doctor to perform one. The negative right to the free exercise of religion is one of the only legal defenses a doctor who opposes abortion has against a law that would otherwise compel him/her to act contrary to his/her religious beliefs.
Lastly, abortion is a much stickier issue in the United States than it is in the rest of the world. This is mostly because religion is a much bigger factor in America than it is elsewhere. Consider the following graph: (Source: Pew Research Center)
Clearly, America is a far more religious country than any other developed nation.
Since the vast majority of abortions in the United States are elective (i.e. not connected to health concerns of the fetus or mother, and not due to rape or incest), the central issue surrounding the abortion debate is a moral argument. Moral arguments against abortion often stem from religious faith, but not always (there are pro-life atheists out there). The only reason women have a right to abortion in America is because of a SCOTUS decision, not an act of Congress. Those are always contentious. Many people in the pro-life camp consider the SCOTUS opinion to be a perverse interpretation of the Constitution, and that the very idea that there is even a "right" to an abortion at all is a matter of serious debate in the U.S.
To understand why the issue is so complex in America, you have to frame the conversation in the context in which both sides understand the issue: (Note that I am using the terms "Pro-Life" and "Pro-Choice" even though different groups call them by different labels)
People on the pro-choice side of the argument see pregnancy as a women's health issue. A woman has a fundamental right to make civil and medical decisions regarding her own body and her own life, and since pregnancy is inherently a medical condition with serious medical and social implications, neither doctors nor the government may withhold treatment to terminate an unwanted or potentially dangerous pregnancy. Extremist pro-choice activists believe that a woman should have the right to make this decision at any time for any reason, while more moderate pro-choicers believe that there should be some limitations. Others may be morally opposed to abortion, but nonetheless believe that the government cannot take away a woman's right to choose.
People on the pro-life side of the argument see a fetus as an unborn child -- a bona-fide human life -- and is therefore entitled to all the rights and protections thereof. Since the right to life is the most fundamental right of all human beings, termination of an unwanted pregnancy is tantamount to murder. Although they recognize that the child's life is intimately connected to health of the mother, the mother has no right to end an unborn child's life for arbitrary reasons. Extremist pro-life activists believe that abortion should be illegal in all cases except where it is medically necessary to save the mother's life (and where there is no hope of saving the child), whereas more moderate pro-lifers would allow exceptions for pregnancies resulting from the rape or incest of the mother.
So you see that in your question, you take it as a matter-of-fact that women have a right to abortion, which is common in many other countries. In America, however, that right itself isn't even a settled issue.