The problem is that there are factors Rod and the others are not taking into account, mostly because they don't know about them:
I think you are not taking several thing into account here.
First, the state constitution to which Davis swore an oath stated very explicitly that marriage was between one man and one woman. Therefore, technically speaking, she would be violating her oath to issue a marriage license to a same-sex couple.
Second, Davis' refusal to issue a license does not prevent anyone from getting married. They can simply go to the next county down the road. In Kentucky this is easy, since there are 120 counties. The couple that brought the suite claimed that that was burdensome, but it wasn't burdensome for them to go to another county to file their suit.
Thirdly, Kentucky just passed a Religious Freedom Restoration Act two years ago that requires the government a) to prove that the government has a compelling interest in burdening someone's religious freedom, but b) that they have to use the least restrictive means in doing so. The judge in this case has ignored that law.
Fourthly, all four legislative leaders here in Kentucky agree that this issue can be easily resolved by updating marriage laws that are out of date anyway, but the governor refuses either to call a special session to deal with it or to issue an executive order relieving Davis of this (arguable) obligation. All he has to do is sign an executive order and everyone gets what they want. But he has, by his inaction, precipitated this crisis.
Finally, this is not an issue of civil disobedience. For one thing, we are talking about a public official, who is, by virtue of being one, part of government itself who claims that she IS following the law. For another, it is not civil disobedience to claim refuge in the Constitution to protect yourself against a misinterpretation of it. You may not agree with that practice, but it is not rise to the level of civil disobedience. You are not refusing to follow the law, but only denying that the "law" you are being asked to follow is really the "law." That is something very different.
Read more here.