I am not going to respond to everyone. What I will say is that this clerk is entitled to believe what she wants and so are those that believe she is wrong.
She is wrong, both legally and morally, and this can be demonstrated through rational argument.
Each person is a moral being created with intrinsic value and worth.
That's not correct. Morality varies by person and culture, and people (animals) aren't supernaturally created, they're born from sexual reproduction. Their worth is only realized by the way they are treated by others.
People are free to believe that marriage is between a man and a woman, or they are free to redefine marriage to mean whatever they want it to mean (like what happened this year).
Marriage was always defined as "whatever they want it to mean." Its implementation in society has changed drastically across cultures and over time. Biblical marriage is nothing like modern marriage.
Both views take a moral stand and make an absolute claim for their beliefs: each side believes that their belief is right.
This is a false equivalence. People that argue for gay marriage make appeals to empathy and the rational, real world effects of having such an institution. Their opponents make false claims and the always ridiculous appeal to divinity.
It would be ludicrous to believe, as moral beings, that everyone is going to agree with everyone else, or that everyone should be forced to believe what the other believes.
What isn't ludicrous is expecting people to defend their moral standards through reason and evidence. You can believe all day, but the real world effect of your actions is the only thing that actually changes people's lives and is thus the only thing that matters.
My person complaint is that I see (as do others) history about to repeat itself. I think Herbert Butterfield's (Regius Professor of History and Vice-Chancellor of the University of Cambridge) analysis on human nature in the aftermath of World War 2 explains the clash of worldviews that this thread and many in the United States espouse and ultimately where it will lead:
Regardless if what a person believes, history bares the scares of what happens when a society declares one side to be morally incorrect and at some point does everything in its power to eliminate those that hold to a contrary point of view. There is just no way this is going to be avoided.
We aren't trying to eliminate anyone. We're trying to make better laws, exercise empathy and convince people that changing their minds is the right thing to do.