The stupidity of that assertion is beyond belief. Same-sex marriage is but one of many classes of actions that are not explicitly forbidden in the written law because there was a history of common-sense common law that would never have considered it to be necessary to prohibit such things. But maybe we can have fun with it. I'd like to get a driver's license for my pet llama. I dare you to find a constitutional amendment, statute or court decision -- that (quote) "explicitly deems void or otherwise expressly forbids a llama from getting a drivers license or from driving a car."
You may object, for example, that the llama will have difficulty passing the written portion of the driver's license test, but with judges such as this one, there should be no difficulty in having the requirement for a written test thrown out because of either the Americans With Disabilities Act (llamas need accomodation for the difficulty of holding writing instruments in their cloven hooves), or because the driver's license test was only able to be administered in English or possibly also in Spanish, but was not available in a language that the llama could understand.
is you llama an american citizen? If not, I don't think you'd have much of a case