Rav Moshe actually held a pro-choice position, even though ruled against abortion in nearly all cases. Since abortion is required if the mother’s life is at stake, he wanted the ultimate decision for whether to allow an abortion to be with the rabbi (who can decide whether the woman’s life is in danger in the terms of halacha) rather than giving that power to the doctor alone or the government.
Remember that the Satmar Rebbes told their chassidim to vote for Hillary last time. Many complex calculations are involved in deciding who is best to elect, so ultimately it must be up to each person.
The Torah is vast and contains a multitude of laws and hashkafic views. How could one candidate or party incorporate all of that? Note that all the religious parties in Israel are left-wing on economic and social-welfare issues but right-wing on security and moral values. There is no equivalent in the US.
And voting is not just a matter of halacha but also a matter of common sense–that is, is the person a stable and reasonable person who is literate, well-read, kind, understanding and deliberates before acting?