In 1973, the US Supreme Court ruled in the case of Roe v. Wade. The Court reasoned that a woman has a right to an abortion under the Constitution’s implied right to privacy. The Court’s reasoning has subsequently been derided by people from both the pro-choice and pro-life movements.
If you want evidence of how disconnected this thought process is, you need look no further than current abortion debates. In my experience, pro-choice advocates rarely, if ever, argue that abortion makes sense under a right to privacy. After all, your right to privacy doesn’t protect you from prosecution should you murder someone in the privacy of your home. If the fetus is a person with equal rights, right to privacy doesn’t begin to cover the right to an abortion. In contrast, if the fetus is not a person with equal rights, right to privacy isn’t even necessary. You don't need to claim right to privacy to legally obtain other medical procedures. Medical treatment is usually involved with issues of bodily integrity.
However, note that Roe itself rejected unrestricted bodily integrity as a basis for abortion:
In fact, it is not clear to us that the claim asserted by some amici that one has an unlimited right to do with one's body as one pleases bears a close relationship to the right of privacy previously articulated in the Court's decisions. The Court has refused to recognize an unlimited right of this kind in the past.
The reasonableness of the right to privacy argument implicitly and explicitly rests on the assumption that the fetus is not a person with equal rights. The Court acknowledged as much:
The appellee and certain amici argue that the fetus is a "person" within the language and meaning of the Fourteenth Amendment. In support of this, they outline at length and in detail the well-known facts of fetal development. If this suggestion of personhood is established, the appellant's case, of course, collapses, for the fetus' right to life would then be guaranteed specifically by the Amendment.
However the Court rejected the assertion that the fetus is encompassed in the term “person” as used in the Fourteenth amendment. Meanwhile the Court did address, separately, the central question as to when “life” begins. The Court essentially refused to answer:
We need not resolve the difficult question of when life begins. When those trained in the respective disciplines of medicine, philosophy, and theology are unable to arrive at any consensus, the judiciary, at this point in the development of man's knowledge, is not in a position to speculate as to the answer.
It is curious that the Court did not feel justified in determining when life begins, but felt justified in determining when it can be extinguished.
In Roe, the Court did grant the state some ability to restrict abortion:
For the stage subsequent to [fetal] viability the State, in promoting its interest in the potentiality of human life, may, if it chooses, regulate, and even proscribe, abortion except where necessary, in appropriate medical judgment, for the preservation of the life or health of the mother.
However this caveat was rendered effectively useless in a companion case decided by the Court the same day: Doe v. Bolton. The Court held that a state may not unduly burden a woman's fundamental right to abortion by prohibiting or substantially limiting access to the means of effectuating her decision.
Doe v. Bolton is of particular interest because of the following quote:
...the medical judgment may be exercised in the light of all factors - physical, emotional, psychological, familial, and the woman's age - relevant to the well-being of the patient. All these factors may relate to health. This allows the attending physician the room he needs....
This quote is important because it implies that "health" exceptions are not just for physical health, but for a wide range of issues. In fact, in this context “health” has become such a subjective term that Doe v. Bolton essentially legalized abortion for a great variety of reasons at any stage of pregnancy.
Since Roe and Doe, there have of course been populous cries for repealing Roe. Some legal minds within the pro-life movement believe it would be best to wait until the Supreme Court's composition is more sympathetic to the pro-life cause. If Roe were overturned, the legality of abortion would be left up to each state.