Why can’t GOP politicians trumpet their religious credentials without assaulting women?  

Because fundamentalist religion of all stripes has degradation of women at its core, and fundamentalist Christianity is no exception. Progressive Christians believe the Bible is a human document, a record of humanity’s multi-millennial struggle to understand what is good and what is God and how to live in moral community with each other. But fundamentalists believe the Bible is the literally perfect word of the Almighty, essentially dictated by God to the writers. To believe that the Bible is the literally perfect word of God is to believe that women are tainted seductresses who must be controlled by men.