Note: The point of this post is not to make the case that abortion is a violation of the NAP, but to simply explain how abortion could be dealt with in a free society, so Libertarians who think abortion is not a violation of the NAP will not like this.
Abortion is a very controversial and emotional issue, both in mainstream politics and within libertarianism. Going against convention, I, a nonreligious, fairly cosmopolitan young person, am morally opposed to abortion, going as far as to say its an initation of force and violates the property rights of unborn children. I do not use the term pro-life, to describe my beliefs, because it, and its corallary, pro-choice, polticize the issue and frame it in terms of a liberal vs conservative, right vs left debate, and as a libertarian, I reject that framework, and the terms that go with it.
With this in mind, I do not support a law banning abortion. I do not think the government could properly enforce a ban on abortion, and would most likely use it as an excuse to violate rights. I instead think that the courts and police should be privatized. Instead of a centralized police force kidnapping people for breaking a law written by a monopoly organization arbitrarily and taken to a centralized court system that has the sole right to declare rulings, I think instead if someone, lets say, James, violates the rights of Samantha, by stealing her sexy sports car. Then Samantha could sue him and take him to a private dispute resolution organization. Lets say the court rules that he indeed stole her sports car, he would have to pay restitution, perhaps returning of the car, plus paying for damages. James would most likely get a bad reputation and be ostracised for his actions, and this would make it very hard for him to do any economic transactions. Infact, if he did enough antisocial behavior like this he would most likely have no where to go, his utilities would be shut off and all the roads, shops, apartments, etc. would shut him out.
Now, to stop abortion through this system, lets say a woman is pregnant, and her lover, has not abandoned her and wants to help raise the child. But she decides she does not want the child and goes and aborts it. The dad can then sue her for murdering his child and collect restitution from her, and she would gain the reputation of a murderer. But what if the mom and dad both did not want the child and decide to abort it, then the grandma could sue on the childs behalf, or the uncle, or anyone within the family. Normally parents posses guardianship rights over their child, but if they do something like abuse their child, or kill their child, then they have given up their guardianship rights, because they are not properly making use of them to take care of the child, so therefore, they would be up for grabs for the next eligible person, which would be a relative. Lets say no one steps up to sue on behalf of the child, then the parents can still be ostracised for their actions, and can become social outcasts.