I think this Kim Davis person is behaving foolishly, as are her Christian and non-Christian supporters, but I would note that the difference between traditional Christianity and both Mrs. Davis' version of I guess what could be called "Civic Christianity" and Islam is that traditionally Christians sought a place for themselves within their societies that did not involve displacing the majority, whereas Mrs. Davis and the Islamic faith both set themselves above whatever law or whatever kind of society is already in place. Hence, for instance, the Syrian Christians in India never set themselves in opposition to the Hindu majority, and only sought their own laws to govern themselves as the separate community they have been since c. 52 AD (and have never been more than a few percent of Indian society at most, so it was wise of them not to do so). Their relative separation helped, in fact, because when the colonial masters came in the form of the Roman Catholic Portuguese, they had a distinct lineage and culture that helped them resist the colonial occupier while keeping their Christianity as it was (Syriac, not Western), via the Coonan Cross Oath. This is likewise how the Copts in Egypt, despite being the numerical majority of society, resisted hundreds of years of pagan government dominance, and later Byzantine (Eastern Roman Christian) dominance, and now Muslim dominance for the past 1400 years. I don't think the likes of Kim Davis or any American Christian could do such a thing, because they are too dependent on what they see as the natural order of things, where they will always be a majority or at least always have the option to exercise their right to practice their faith via whatever means they wish (even when that includes citing Jesus as a reason to not do their jobs). I don't know how it could so consistently escape the view of these people, but that's not the way that Western societies have been headed for the past 500 years or so, and the secularization is only increasing, and can't be stopped by simply pretending that you can function within it without having it change you, either by force or by persuasion.
So it is better, I think, to try to function as our fathers have in Egypt, Syria, Iraq, and across the Middle Eastern homeland of our faith, where we have been minorities for most of the last 2000 years but always maintained our positions by separating ourselves to some degree from civil society. After all, in places where we are seriously imperiled now, like Iraq and Syria, we are in such a state because "Christian" leaders order other "Christians" to go to these places and destabilize the governments there for the sake of bringing liberty to the non-Christian majorities who use that liberty as a means to further crush us, so what can a Christian say about law protecting his or her faith? That's almost never been the case. Even being part of the Eastern Roman empire, no Coptic person experienced peace by having Byzantine rulers thrust upon them, even though these were fellow Christians in some sense. We need to stop being naive and taking things for granted and start being smarter and working small to do things that won't get on the news, but will register very consistently how we wish to live. Read: Clean our own houses first, don't hide our religion in business (to the extent that it makes sense to bring your religion into your business, that is; I don't think anyone cares if the garbage man prays to Allah or Buddha or whoever so long as the garbage is picked up), etc. Don't be these lazy, scared wimps or pompous holier-than-thou types. Our fathers fled into the deserts of Egypt, Syria, Palestine, and Asia Minor in times when persecution was much worse than someone being fired from issuing licenses she didn't want to issue in the first place, all for the sake of their religion and being able to live as they wish. And it didn't cost the society anything to have people who would rather pray living in caves and cells, but it did eventually attract people of similar mindset until a truly Christian society flourished for the ~4 centuries between the days of St. Anthony the Father of the Monks and the coming of the Arabs, who realigned society in their image in a way that is actually very similar to what both Mrs. Davis and the aggressive secularists are trying to do. The only difference is that Mrs. Davis is much less comprehensive in her application of religion to her work in the public sphere than Islam is, as Islamic law does not recognize any separation of religion and state. I believe it is better, knowing this, that Christians and others concerned with the preservation of the United States constitution, which contains within it an article guaranteeing freedom of religion, use their energies to guarantee that the Separation of Church and State so often touted by the secularists be preserved with equal force in both directions. In other words, yes, this woman's office is not a church, but free exercise of religion ought not be limited to specific buildings if you can successfully show that it isn't by matter of design, and not just whatever your personal affinity for particular parts of its laws compels you to do (i.e., you don't get to take over the common lunch room at work to hold a service, but you also should not be disallowed from praying at work, should you follow a religion that has hourly prayer at specific times, like Orthodox Christianity, Islam, and Judaism do; many companies, universities, etc. now have washrooms for their Muslim populations, which require ritual cleaning before their prayers; if Christians in the USA were smarter, we'd be pushing for our own religious rights in similar environments, rather than to take away others' rights as this Kim Davis did). That's the problem with the Kim Davis's [and Islamists] of the world: She wants to make her exercise of religion, which she is entitled to, something that everyone must follow. And her exercise of that religion includes turning away gays who came to her office as an alternative to getting married in churches in the first place. I don't agree with gay marriage even slightly, but if I were gay I would not be pleased with that idea.
So I don't think it's a zero-sum game, as plenty of people have tried to make it: "Oh, so you're with the secularists who would rather legislate us out of existence?! For the sake of the gays, who would do the same?" Nope. I'm with people recognizing that if we do not want to feel the brunt of anti-Christian and broader anti-religious sentiment of people who will one day be in positions of power in America and remembering Mrs. Davis unfavorably in deciding how to interpret and enforce laws and the constitution (and hence tarring us all with the same broad 'bigot' brush as an excuse to take away the rights that we do have, as though the Coptic Orthodox Christian, the Syriac Orthodox Christian, the Armenian Christian, the Greek Orthodox Christian, the Roman Catholic Christian, etc. are all in 'cahoots' with the Evangelical Protestant likes of Mrs. Davis by virtue of all being Christian), we ought to be a lot smarter in how we approach questions of our churches' and faith's interaction with government. Where we can improve the visibility and lot of our people, we ought to do that (e.g., why are Muslim organizations in America successfully getting content about their religion put into public schools as part of history courses and 'diversity' programs, but no Eastern Christians have done the same? This is dooming an entire generation to grow up ignorant of the real facts and history of the region, and presenting Islam uncritically as the native religion of the Middle East and a very positive force in a way that never happens with Christianity, Judaism, Yezidism, or any other Middle Eastern religion...if you're an Eastern Christian, you ought to care about this). But being self-righteous and demanding as though we are in the position to force our religion on any given society by fiat is not the way to go.
Of course I believe Christianity is the only 100% true faith and the religion that everyone should be a part of, but it has simply not been the case historically nor currently that this impulse is successfully married with state law or state-sanctioned force without suffering as a result. And not only does our faith suffer, but then people suffer at the hands of whoever is (en)forcing it, too. Why on earth would we want to so poison our Christianity in the minds of the rest of the world, thereby bringing shame upon the name of Christ our God, so that some yahoo in Kentucky doesn't have to do her job when the laws change in a way that contradict her newly-discovered beliefs? Are we so beholden to laws and having a favorable environment that we cannot function without it? It's odd, when you consider that Christians in a lot of the rest of the world don't depend on that (because they live in places where the law code favors others, and the society too), yet they provide the strongest witness to the truth and love of Christ.