In he United States, abortion became legal in every state by order of the United States Supreme Court in its 1973 Roe v. Wade decision. In 2022, in the Dobbs case, the Supreme Court returned most abortion regulation to the individual states, which left each jurisdiction to deal with the two issues that the Roe court had never fully resolved.
The first is the issue of whether the child is a living human being with all the rights that go with that status. The second issue is the right of a woman to do what she wishes with her own body—sometimes characterized as a privacy right—but one which libertarians would more likely consider a property right.

The Court had never seen either issue as absolute. Instead, it found that the fetus did have some rights that states could protect, but that the mother’s rights trumped her unborn child’s rights until the baby became “viable” outside the womb. It is also worth noting that the courts have never recognized a women’s right to kill the child, but only her right to end her pregnancy.
Pro-choice libertarians appeal to the same two issues in arguing that no one may prevent a woman from obtaining an abortion. As to the first issue, the question is whether the unborn child is a human person. If the unborn child is not a human being, then it has no rights to protect; no rights even to weigh against the mother’s rights. Denying the humanity of the unborn immediately short-circuits the analysis and ends the debate for those pro-choice libertarians wishing to avoid confronting the non-aggression principle.
Thoughtful pro-choice libertarians, however, will sometimes concede—for argument’s sake—that the unborn child is a person, and then proceed to justify abortion on libertarian principles. They claim to base their arguments on the non-aggression principle—and if their arguments were valid, we would have to concede that the pro-life position (and Catholic teaching) is incompatible with the non-aggression principle and—by extension—libertarianism itself. Murray Rothbard was one who recognized the need for a libertarian analysis rather than a mere dismissal of pro-life claims.
Murray Rothbard on the need for a libertarian analysis

Rothbard argued the pro-choice position in his books and articles, notably in his libertarian manifesto, For a New Liberty (1973). He does, however, begin with a fair consideration of what—at that time—was chiefly the position of Catholics:
“For the libertarian, the “Catholic” case against abortion, even if finally rejected as invalid, cannot be dismissed out of hand. For the essence of that case—not really “Catholic” at all in a theological sense—is that abortion destroys a human life and is therefore murder, and hence cannot be condoned. More than that, if abortion is truly murder, then a Catholic—or any other person who shares this view—cannot just shrug his shoulders and say that “Catholic” views should not be imposed upon non-Catholics. Murder is not an expression of religious preference; no sect, in the name of “freedom of religion,” can or should get away with committing murder with the plea that its religion may (or may not) permit it. The vital question then becomes: Should abortion be considered as murder?“