Let's be perfectly clear on one thing. This whole argument about abortion is not about whether a fetus is alive or not. It's about whether a person has a right to end that life. Medical science is clear about the first question. It's alive. Criminal law is also clear about the second. It's murder. There should be no debate about this.
And yet there is. That's usually an indication that there's disagreement over the premises of your argument. In this case you seem to be implying that 'alive' is synonymous with 'legally recognized person'. I'm not completely sure on where the legal statutes are on this, but there's clearly no consensus on when a fetus becomes a person. Some people think it's not a legally separate person, with distinct rights, until it's separated from the mother - ie 'born'. Other's think that from the moment of conception until birth, we are facing a situation of one person living inside another.
There's also a question of jurisdiction. I don't think government should be granted authority over the inner workings of a person's body, regardless of the excuse. Any kind of coherent picture of individual rights puts "self-ownership" front and center. Without that, as a basis, we have no rights at all.
I think Republicans are indulging the same conceit liberals are usually accused of: they think every problem in society can be solved by passing a law. And no matter how many times it's shown to them that law isn't, usually, the right answer - they are convinced that THIS time it's different.