The woman is the Kern County Auditor-Controller-County Clerk, an elected position. She cannot be fired. She actually tried to resign as County Clerk but keep her position as Auditor-Controller. Apparently, she couldn't.

She is not required by law to perform marriages, so she would only be violating the law if she refused to perform same-sex marriages but continued to perform other marriages.

This is an interesting response to the County Clerk's action:

Local same sex marriage advocate Whitney Weddell isn’t overly concerned that Barnett’s office won’t offer civil ceremonies. “Who really wants to be married by a hater?” she quipped. What the gay, lesbian and transgender community has always wanted, she said, is the marriage license. Weddell said she actually respects Barnett’s attempt to step down as county clerk. “I respect her attempt to abide by the tenets of her faith,” she said.

I don't agree with Weddell's use of the word "hater" for someone that she later contends is "attempting to abide by the tenets of her faith."