Nate Silver devotes an entire article to the fact that states that ban gay marriage have higher divorce rates, and says he thinks it's interesting.
Some possibilities I thought of:
-States with high divorce rates tend to be poorer (because poor couples are more likely to divorce, either because of the money issue or because they tend to marry younger), which corresponds with more religious regions of the country, either by coincidence or cause (I suspect cause but I have little evidence, so I'll accept either explanation).
-Getting married as a gay person is harder, so those who do it are more committed and more likely to stay together (small contributor at best)
-People in unhappy marriages are less happy in general, and are stronger in their religion (thus banning gay marriage) or simply becoming bitter (and hating those who are different, like gay people)
I tend towards the [edit: FIRST] explanation. [I changed the order].
Another possibility, of course, is that it's a coincidence the same as the link between number of pirates in the world and global warming. As the number of pirates have gone down over the past few centuries, the earth has gotten warmer! (and if the world has in fact gotten a bit cooler in the past few years (due to a megacycle or a temporary fluctuation), the temperature has dropped as Somali pirates have risen in number!)
If Nate Silver were to be reading this post, I'd caution against thinking that statistical relations imply anything even remotely interesting without some sort of counterintuitive explanation for it. This is going to be intrinsically related to religion, with no surprising results.