She has a bureaucratic job to do as a clerk which she refuses to do, including signing marriage licenses for straits. It doesn't mean her signature gives a blessing or any sort of religious approval by her personally.
There is only one legitimate justification for her defiance, and it is possibly a big one, that is that the supreme court may have stepped beyond its constitutional jurisdiction in forcing gay marriage as the law of the land. So, her argument should be she defies the legitimacy of the supreme court's decision and hence continue to act according the the written law and disregard the supreme court instruction. It has nothing to do with religion.
It is common for Xtians to identify entirely with their religion. Having been in cults through everything in the middle to Jesuit school, on cults they're incapable of seeing themselves as anything but a "member", a tool in the system, and even the most intellectual and philosophical priest or parishioner will identify as Catholic first, a person second (in name only Catholics with no actual belief involved are a complicated exception, with different levels). This is hard to understand from a Jewish perspective (outside of certain "closest cult to Judaism" divergent cults) as mitzvot and practicing the religion is involved with virtually every aspect of life, but the difference is that since there's nothing to know for certain, there's no way to prove intellectually that they are doing the right thing (excluding deeds that are congruent with Torah, not that they will likely learn these intellectual justifications), so they have to embody their religion, and what is typically told to new missionaries before setting out to hunt is "let them see the love of Beesus inside of you" for cults, to cause lonely and broken people to desire to be less sad, and be drawn to them because they don't look like they are, and more mainstream groups follow "by their fruits you shall know them" as a missionizing tactic, being that standing for their religion shows strength, and since many of their attitudes are taken from Torah, and therefore are beneficial, they believe that others will be drawn to them through this.
Obviously, she made a political argument she put under the mantle of "because of my religion". This is tactical, I've seen it done before as a way of playing on fellow believers' loyalty to the religion, so they might stand with her, but primarily (besides the fact that if she says she does not recognize the authority of the supreme court, the supreme court won't be very nice, claiming religious freedom gives her something she can argue in court, ironically) it serves as a moral justification for the deed; "the supreme court has done all these things illegally, but because of my morals, this is where i take a stand".
Source where she won't issue license to normal couples. As a Jew, you know that signing such a document would mean you have a hand in it. I personally thought "why doesn't she resign" first, but it's the supreme court doing the illegal action, they're the ones that should be forced out. There is no change unless you take a stand. I personally talked an American who Xtian on the fence about the candidates, and he told me this decision by Ted Cruz to get her out of lockup has made him and his family agree with my assertion that he is the best candidate, and they'll be supporting him.
If every clerk refused to do it, and others with "bureaucratic jobs" refused to sign a paper to be handed to Hashem that reads "destroy here", America could fight off this illegal action. So now to get back to my first point a stronger argument than the fact that it violates the law of America, though that's the one you're claiming is OK, is that it violates the laws of heaven. I'm not a posek to decide if signing a paper does anything, but I read the article that said the degenerates that wanted to "marry" were unable to because of one resilient clerk. She stopped American tax payers from proving money to pay for someone to engage in sinful acts, even if only for a short while. Maybe she should be fired from her job, someone brought up the argument of the tree humper who didn't want to issue conceal-carry permits, and while the argument has no bearing, since the clerk can only use as backing that he feels that's what right, he stands against the law, the court and G-d didn't leave any books around backing him, still I think a person that refuses to issue a conceal carry permit should be fired, and if this person refused to issue a licence to a straight person, they should be fired. If they live in a country like Canada where the majority does support paying homosexuals to sodomize each other, then she shouldn't take that job (that'd be outta the stage 1 jihad playbook) and if she had done this, she should be fired, which would be for her own good anyways.
She has taken a stand and said "i don't have to give up my job or my America, I want to do what is right and not sign my country's death warrant to Hashem (I know, but still better someone who "fears" his statue than someone who "fears" nothing), I believe that we can live in a country where government workers are not forced to violate G-d's laws (and she should rightfully add, the constitution) and I can do what is right without having to run from evil, I take a stand on this".
I don't care if she's a democrat, and I don't care what religion she said it for. She's being a better human than the clerks that are signing away Americans to a life of supporting degeneracy, and their country to destruction because they were "just doing their jobs".
Ted Cruz told entire states to ignore the supreme courts illegal decision and not to let them force them do this. Everyone should oppose this in any way possible in their daily lives. This woman is an example that America still has what it takes to fight for what is right. It does have to do with religion, just not the way she said it, but if she was a pastafarian, invented a verse of the magical unicorn that said she can't do that, and used that as her justification, I'd still side with her, because she already has the support of the law and the American people behind her, and we can get to all the things that are wrong with her and everyone and thing else in the solar system after we congratulate her for doing the right thing. Her religion argument is not "because i want to make my religion reign supreme, it's "you're forcing me to be a part of something that me or no one else wants or agrees with, and I won't give up my job or turn my job into an evil position".
If you were the clerk in a bureaucratic job, and a bill from Obama just passed with a 2/3 congress majority backing it, would you sign the Iran nuclear deal into existence, if you could stop it, or at least delay it while they're figuring out how to best humiliate you before throwing you in jail? You don't get the defense of the law, just Hashem on this one, would you morally justify your action and say it is right not to do this and what would you say to another man that does it and uses Judaism to back his decision?
The Iranian nuclear deal and everything in the first place is all in G-d's control and will. If he doesn't think America should get nuked, America won't get nuked. G-d bless him, but that action does more to stop a nuclear holocaust than 10 perfect Ted Cruz speeches on Iran. Get a society that can stand up to sneaky judges, you might get one that can stand to sneaky Arabs that control the Persians.