Does the abortion debate infuse religion into government which are supposed to be separate?

Yes, the abortion debate is largely framed on religious grounds. However, religious individuals are allowed to have a voice just as non religious individuals are allowed. The separation of church and state is about keeping the organizations of church and government separate. There are churches that support abortion rights as well as churches that obviously oppose it, so it’s not like there’s a single religious organization that’s solely responsible for spearheading the pro life movement.