Since the SCOTUS ruling there have been a bunch of reactions from conservative religious folk. I have no problem with someone believing whatever they want and I fully expect that when someone has a deeply held faith commitment, they will act accordingly. If they do not, then they are simply hypocrites.
So here are the two problem areas... First, I hear folk, especially very conservative Christians, claiming that the government will force them to marry gay couples or they will somehow be punished. My first question would be when has the government ever forced you to marry anyone? If a couple comes to me as a clergy person, I decide whether or not to perform the service. If not, it may be because I do not agree the couple is ready for marriage or any other reason that I decide. If the congregation I serve has criteria to which I've agreed, then those criteria help shape the decision. If I served a congregation that refused to perform services for divorced persons and a couple came with one of them being divorced, the service would not be performed.
This simply is not a concern and there is no reason to believe it ever will be. Add the question of what gay couple would want their service to be performed by someone who believes their relationship is an abomination and I really cannot imagine it ever becoming a problem.
The second is a bit trickier. We've seen already that in some states, clerks are being advised that they do not have to provide marriage licenses to gay couples. This is problematic in a number of ways. First I would ask whether those same clerks are allowed to refuse a license to a couple with a divorced person? Or a mixed race couple? Or a couple of mixed religions... or no religion for that matter (if indeed, marriage is an institution established by God)? All these situations trigger religious objections by some.
Then there is the question raised by the recent meme... if giving a marriage license means that you've participated in the gay marriage, does selling a gun mean that you've participated in a murder committed with that gun? Does giving a couple a license imply anything other than that the state allows this relationship and that the employee is acting as an agent of the state (his or her job).
Years ago I had a discussion with someone who worked for a defense company and had been condemned by some for working at a plant that produced munitions, implying that they were responsible for how those munitions were being used. His answer was, "I didn't send the troops there. I didn't authorize the expenditures. You elected that government and you paid for those munitions. Maybe you're the one who is responsible." The argument made sense. I think this is a similar situation. If an individual clerk cannot fulfill his or her job, they should find a new job. In doing that job - providing a marriage license - they are not participating in the acts that follow.
On the other hand, what about a JP or Judge who performs weddings as a part of their job? That person is participating in the ritual in a more significant way and religious objections may have a real part here. At the same time, the local government is required by law to provide that service. I would argue that in those circumstances a religious exemption might make sense but that the government agency is obligated to provide someone who will perform the ceremony. John Smith may indeed say, "I cannot do this service for religious reasons, but my colleague, Mary Jones will. Let me get her." Or indeed, while talking to the couple, the clerk need not even mention John Smith. The clerk can just make the appointment with Mary Jones. Under no circumstances can the county office say, "Sorry, we don't have anyone to do your service." They are responsible to provide that service to all who are legally entitled to it