First I want to thank
@SkepticThinker for pointing out
@ImmortalFlame 's post addressing this because I remember it being one of those head slapping moments when I read through this thread earlier.
This is a whole lot of "no". If you find yourself in a situation where someone is doing something which makes you uncomfortable, and the thing they are doing is completely legal for them to be doing and they have every right to do so, then it is not them that need removed from the situation, but you. If you find yourself in a designated smoking area and there is someone smoking there and you don't like the smoke the YOU leave that area. You do not make them put out their cigarette. In a public setting, where the public has rights, if you don't like any part of the public exercising those rights then YOU leave. Not them.
Likewise, if someone works in the public sector, their wages paid by tax money, then they work for ALL the public whether they religiously agree with anyone or not. If a person cannot separate their beliefs from the job they are to do then perhaps they should not have that job. That goes for not just judges and country clerks and the like, but also for businesses who intend to make their money in the public sector. In businesses which deal with public accommodation one cannot simply refuse service to someone because they are black, or because they are female, or because they are...gay. When one takes this absurd idea that it is okay to discriminate against homosexuals if your religion makes you think that you don't have to "deal with" them just how far does that go? Providing car insurance? Medical care? Adoption services? Or do you simply draw the line at businesses which deal in the business of matrimony? Discrimination is discrimination and if you go into a job or business in which you are going to have to be dealing with the public then that means ALL the public. If a person can't handle that without discriminating against people then perhaps they should work in such lines of work.