I think if someone is getting a judge to officiate a wedding, they're not doing it in a ceremony, but in a perfunctory way, at the courthouse. They literally just want to make the union legal. Which it is, even in Texas, so this judge has no standing to refuse.
I thought this already came up when that woman in Kentucky refused to sign marriage licenses for gay people, and it was ruled that while she didn't have to personally sign it, if she refused it was up to her to find someone to do it instead of her.