With respect to the State's important and legitimate interest in potential life, the "compelling" point is at viability.
This is so because the fetus then presumably has the capability of meaningful life outside the mother's womb. State regulation protective of fetal life after viability thus has both logical and biological justifications. If the State is interested in protecting fetal life after viability, it may go so far as to proscribe abortion during that period, except when it is necessary to preserve the life or health of the mother.
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OUTLINE OF KEY POINTS
Roe v. Wade is a LEGAL ruling, not a moral analysis. It deals with the legal question of whether abortion, as a medical procedure, is constitutionally protected. Roe. v. Wade argues that
in the absence of explicit legal rights for fetuses, and with a Constitutional implication against the existence of such rights, the court (and state legislatures!) had no reason to treat abortion differently from other health procedures, particularly contraception.
Three key points:
A fetus in not a PERSON under U.S. law.
Persons have rights under the Constitution, and it is clear that the authors of the Constitution and its amendments did not regard fetuses as persons.
In order to say that fetuses are persons under U.S. law, the Constitution would have to be amended to say so. Therefore the intentional killing of a fetus does not have same legal status as the killing of a person.
States can create laws to protect citizens from harmful practices, and it can ban medical procedures that are harmful. When abortion was initially banned by most states, it was a dangerous procedure. Medically, it is now safer than childbirth. Therefore there is no longer a good reason for states to ban it as a medical practice.
Since 1891, the U.S. has recognized a right to privacy in some "zones" of activity, which means that individuals can make decisions and act upon them without informing other people and without state interference. (Example: Your discussions with your lawyer are private and confidential.) The court has previously recognized that adult women have a privacy right when it comes to contraception and reproduction.
Conclusion: Because fetuses are not legally protected and abortion is a safe medical procedure protected by privacy rights, adult women have the right to receive an abortion in the first six months of pregnancy, and states can only interfere where the interference is appropriate to the woman's health.