Clerks are employees of the government. The government in the United States provides freedom of religious beliefs in the very first amendment of the Constitution. We have now entered a time when "religious beliefs" cannot interfere with what one does. What, then, when one's religious beliefs command them to do something? Then, the ideologies of today are diametrically opposed to the Constitution and the basis of our country. We simply cannot hold modern ideology on a higher level than the basis of our country's foundation.
Your job is not your religious organization. If you are required to do things in your job that do not align with your religion, accept the compromise and do your job, or find a new job.
It is not the gay couple's fault that you took a job that is inconsistent with your beliefs. The law of your faith is not the law of the land, nor should it be.
If you were Muslim, would you work on a pig farm, packing meat?
Now, if your employer agrees that they will give such cases to other clerks as a reasonable accommodation for your beliefs, then good on them, they are a good employer, and are fulfilling their legal function without having to make you directly violate your ignorant religion. At the same time, if no reasonable accommodation can be made, then you should likely be fired, as your exercise of your faith is depriving another person of their rights.
No, clerks should not be allowed to cite religious beliefs when they deny same-sex couples marriage licenses. One reason why they have laws regarding marriage is so that no one has to c are about the clerk's individual beliefs. If the clerk cannot do his or her job properly due to his or her religious beliefs, then the clerk should quit.
Citing religious beliefs as an excuse not to grant marriage licenses to same-sex couples is ridiculous, thinly veiled bigotry. Clerks cannot pick and choose who gets to marry, as their job requires them to be completely impartial in that very situation. If they don't want to grant marriage licenses to same sex couples, then they should not have accepted a job which requires them to do exactly that. It's like if someone took a job making sandwiches, but refused to make any sandwiches with wheat bread because they don't like wheat bread; it doesn't matter if you don't like it, your job requires you to deal with it by definition, and you won't be the one eating it.
Government and religion don't mix and never really have. If these clerks feel so strongly than they need to resign from their jobs not stand there getting up on a soapbox violating the laws they swore abide by in doing the job. These are public servants who must abide by what laws are passed, if their religious beliefs tell them not to than it's time to get out because they aren't doing their job.
Clerks are agents of the government and cannot discriminate against anyone; because everyone is entitled to equal protection under the law. Therefore, clerks cannot cite religious beliefs in order to deny same-sex couples marriage licenses. Owners of private businesses should have the right to refuse services that object to their religious beliefs; but not agents of the government.