The theory that allowing gay and lesbian couples to marry will somehow “undermine” heterosexual couples’ marriages is almost too silly to investigate. But for the record, Nate Silver determines that banning gay marriage is statistically associated with elevated divorce rates:
The differences are highly statistically significant. Nevertheless, they do not necessarily imply causation. The decision to ban same-sex marriage does not occur randomly throughout the states, but instead is strongly correlated with other factors, such as religiosity and political ideology, which we have made no attempt to account for. Nor do we know in which way the causal arrow might point. It could be that voters who have more marital problems of their own are more inclined to deny the right of marriage to same-sex couples.
At any rate, one seriously doubts that this sort of research would be persuasive to anyone. Certainly if the sign on the coefficient of correlation had gone the other way, I wouldn’t suddenly become an opponent of marriage equality. But if anyone you know seems genuinely concerned about this aspect of the issue, please urge them to check out Silver’s table.