The spectacle of Congress trying to intervene on a case-by-case basis in these types of personal decisions is something which turns off most people. It'd be one thing if 6 months ago they'd try to enact some sort of general legislation about these types of issues, but watching them leap into action at the last minute and try to poke their noses into a family's choice that all of us hope we never have to make is going to disturb a lot of people.
As for the Democrats -- not all manufactured culture war events require you to play for the wrong side. People don't always side with the Republicans and their ridiculous spectacles.
...and, here are some actual poll results.