For those of you who are interested in the actual legal aspects of the Kim Davis case (rather than their political or moral aspects), Eugene Volokh gives a very solid and detailed analysis of the laws in play here, and more generally how religious objections are handled under US employment law.
The technical details here get pretty hairy, but the short version is that there's a resolution which will likely work, if Davis files suit in state court (the case against her being in Federal court) to request a declaration that the county may issue marriage licenses without her name on them (which is apparently the actual accommodation she wants, one which is almost certainly a fair request under Kentucky state law), instead either putting a deputy clerk's name or simply "Rowan County Clerk." That's different from the question of letting her deputies issue the forms, since they're still issuing forms with her name on them, which is apparently what she objects to, and she can't simply make this change on her own, because without that court order the resulting forms wouldn't be legal. But with that court order, the forms are legal, any of her deputies can issue licenses, she's apparently satisfied as well, and that should satisfy the Federal judge, so we can all go home.
If that last paragraph seemed confusing, it's not half as messy as the actual legal details come out to be. But the key point in it is that this case has been eaten alive by politics and the press, and there's actually a resolution that would make everyone happy. Davis isn't, apparently, actually demanding that no gay couples be allowed to marry in her county.
If it seems to you like I'm deliberately sidestepping the point, I'm actually highlighting a more subtle point, which is that these kinds of disputes have been a feature of American law (and European and Islamic law before that) for centuries, and we actually have a fairly well-structured system for resolving them. Generally, people can ask for "reasonable accommodations" for their religious beliefs, and reasonable does actually mean reasonable – it's defined in terms of the burden it would put on the person's employer, colleagues, customers, and so on. For example, courts have repeatedly ruled that "if you want [X], go to another town" is not a reasonable accommodation that people should have to make.
This is important, therefore, not just in this case, but in the much broader category of cases where individual beliefs clash with job responsibilities. This ranges from the vegetarian bus driver who didn't want to hand out hamburger coupons to the evangelical pharmacist who doesn't want to dispense birth control pills. The law already has a meaningful balancing test in it: the question is how best to apply it.
One important point which Volokh's article doesn't bring up directly is the ways in which different services will have different bars on reasonability. To take an extreme example, there's basically no kind of patient an EMT could reasonably refuse to treat in the field. A doctor may be more or less limited, depending on the ability of other equivalently capable doctors. The areas where this is most important are areas where there is a de facto monopoly on provision of services, with government services (which are a de jure monopoly) being the extreme case. Another point which this article doesn't mention (but which is relevant in other cases) is that in some states, certain bases for discrimination are per se illegal, with religious reason or not – but that varies a lot from state to state.
In practice, there are three kinds of cases involving this law which are of great importance right now.
The first involves pharmacists who refuse to dispense certain medicines, e.g. birth control. The key issue here is whether those medicines are treated as medical necessities by the courts or not; if a judge decides that this is a type of luxury, then a customer not being able to get it might be considered a far more reasonable accommodation than if it is treated like an ordinary medicine.
The second involves health insurance, which under US law is mostly provided via employers. The subtle issue here, which many courts haven't touched, is that when a business does act as a provider of health insurance, it's doing so under sufficiently strict guidelines from Federal law (especially post-ACA) that it could be argued that it's acting not on its own behalf, but as an agent of the government. If that's the case, then the underlying service to be provided is the entire package of the insurance, and that provision is itself legally mandated; the business may then ask for reasonable accommodation in how that is provided, but not necessarily in what is provided. (This is an argument I was rather surprised didn't come up in Hobby Lobby)
The third case are the "cake baker" cases, which basically boil down to a business' right to discriminate among clients. This is actually a far more complex and subtle issue than it at first seems, and it probably merits an entire post of its own, so I won't start down that path here.