I don't think any magic occurs at birth, or at any particular stage. But, I think the argument comes down to 2 competing interests: the mother's control over her own body, and the fetus's human rights. Ultimately, the decision is whether a fetus at every single stage should receive the same human rights as people already born into this world.
As I've stated on the thread a few times, I think considerations like viability, cognizance, brain & nervous development all contribute to what we ascribe to "personhood," and are integral to a discussion of actual human rights. Any point after conception will necessarily be arbitrary, but I think there is a point up through which most would allot that a woman could make a decision to abort the fetus without what we would call "murder" taking place. To me, that's an appeal to emotion, and nothing more.