Protecting the rights of a part of a woman's body that is not a separate, conscious organism (while it certainly has a nervous system and can feel pain and discomfort, it is, for the most part, a non-sentient extension of the woman herself until late in the pregnancy) is quite different from protecting the right of an actual living, breathing, separate entity.
The only argument that matters in the abortion issue lies in when the fetus can be considered a viable human being, thus gaining human rights. Until it can be considered human, an embryo or fetus is merely a collection of cells growing inside the mother's body.
And having said that... and not knowing anything about the legal cut-off date for abortion in the US, I've just looked it up, and that's how the law stands in most states, wherein abortion is only allowed until a certain number of weeks, typically between 20-28 weeks, based, apparently, on when the fetus becomes "viable".
The record for the most premature birth survival is around 21 weeks, while full term is around 40 weeks. At 30 weeks, most fetuses can survive premature birth and already have the capacity to develop consciousness.
As to why consciousness is not a determinant in the cut-off, research shows that consciousness may not develop until a few months after birth. And abortion after birth would simply be murder, as the child is legally recognizable as a separate organism, and a human, and thus possesses human rights.