Hold on, now. Are you suggesting Christians may no longer voice their views publicly on this issue? Even the Supreme Court backed down from that sort of militantism in Obergefell:
"Finally, it must be emphasized that religions, and those who adhere to religious doctrines, may continue to advocate with utmost, sincere conviction that, by divine precepts, same-sex marriage should not be condoned. The First Amendment ensures that religious organizations and persons are given proper protection as they seek to teach the principles that are so fulfilling and so central to their lives and faiths, and to their own deep aspirations to continue the family structure they have long revered. The same is true of those who oppose same-sex marriage for other reasons.”
So we know that she is certainly allowed to speak her mind. More than that, we know that Civil Disobedience has always been part of the American legal system. Lincoln was the most high-profile offender, others like Rosa Parks and MLK did similar things.
Look, it’s not that she’s refusing to give marriage licenses to LGBT folks; she’s refusing all marriage certificates until the state permits her to remove her name from marriage certificates. This would require… what? Maybe fixing a broken state statute. So why is it out of the question for the State to fix a broken statute? Especially when the alternative is barring those who disagree with gay marriage from that public office. That’s like making voters take a “literacy test” at the polls. Or making people swear allegiance to the party and its Fuhrer before holding office.