Attention: biology lesson here folks:
Sperms and eggs contain half the DNA of a person. Neither a sperm nor an egg are a "potential person", but rather specialized cells. While both are living cells, neither of them are living organisms. Neither is capable of self-reproduction at a cellular level nor reproducing new organisms.
Once a sperm and egg come together, the new entity (known medically as a zygote) has a full complement of DNA, is capable of cellular division and more importantly can grown and will eventually become a fetus and then a baby, a teenager and an adult. At the point of conception, the fertilized egg becomes a unique member of the human species.
All of the above is undisputed scientific fact.
So now the question becomes one of rights. Does the the principle of self-ownership apply to unborn humans? If there is any doubt about whether this human has a "spirit" should we error on the side of protecting life or killing it?
Just because a fetus is not visible to me does not change the objective reality that he or she is member of the human species. While the death of a fetus may not feel as important to you as the death of an adult you know or might read about on the news, the objective reality in either case is that a human dies. Why should the humans we see be more protected than the ones we don't see?
The measure of a culture is how it protects it's most vulnerable members.
If a culture can promote the killing of completely innocent unborn children, then there is really nothing left that won't eventually be done.
It is no coincidence that the legalization of abortion has preceded many of the other social ills we face today.