If they [rights] do not come from God, then they are simply a social construct...
This is where you are wrong. There are formulations of rights which are neither mere social constructs nor based on religion—which is, in the end, just another variety of social construct. My preference is the one based on the legal concept of estoppel, which can be summarized as the logical principle that one cannot rely on incompatible claims within the same argument. For example, one cannot consistently argue that one has the right to act in a certain way toward others while simultaneously claiming that those affected lack the right to reciprocate. Either everyone has the right or no one does. If the right exists then the first party infringed on it and deserves the punishment; if not, then neither the original action nor the response infringes on anyone's rights.
In this case there is the additional complication that "the right to marry" is really referring to a number of different aspects of the law, not simply the right to hold a marriage ceremony and consider oneself married but also power of attorney, visitation rights, joint taxation, common ownership of property, etc. However, the gender of the two parties is irrelevant to all of these legal considerations; there is no reason whatsoever that the law should permit e.g. visitation rights to a couple composed of a male and a female, but deny them to a couple composed of two males or two females.
If certain individuals of a religious persuasion wish to consider homosexuality a sin, fine. They don't have to practice it themselves, or even associate with those who do. But there is certainly nothing in the Bible which would require anyone to deny that the relationship exists, or to refuse such couples equal rights under the law. This ruling is about the law, not religion.