this is anecdotal evidence, but in my experience, men tend to be more in favor of abortion than women... because it's a way to escape consequences and financial ruin for them. (this is true for women too, obviously. kids ARE expensive!)
but the most savage and relentless opponents of abortion that I've known over the years have all been women. it seems like a lot of women probably build their identity around the ability to give birth... like they consider it some sort of sacred gift from Nature and think that all women should hold pregnancy and children in high regard. there isn't some universal sisterhood of women that is always out there fighting for women's rights.
in a similar vein; it's like that proverb that "slut-shaming is something men do to women"... when, in my observations, body-shaming and slut-shaming against women is perpetrated by other women the vast majority of the time. it's probably based on the different methods that women use to compete against each other. women will say truly vile and vicious things about each other, and to each other, that most men couldn't even imagine.
I guess where I'm going with this is... if you assume that all of this culture war stuff is being done by conservative white evangelical Christian men... then you probably don't truly understand the people you're fighting against. [this is, of course, assuming that you're actively trying to stop them]