I don’t know what the Democrats are going to do, but this seems a safe prediction. I’ll also toss out what Republicans will likely do: Run away from this decision as fast as they can. They’ll not tout the religious exemption, they’ll not defend the decision, they’ll try their hardest not to mention it at all, thus hurting THEM in the midterms by making them seem what they are - weak kneed liberals that just don’t have the guts to call themselves Democrats.
You called it just right. I suspect some of them will also add a few words of support for gay marriage.