An increasing number of jurisdictions are now making same-sex marriages official. Looking at the United States, it would appear that more liberal states are making this move, while more conservative ones move in the direction of banning them. Arthur Lupia, Yanna Krupnikov, Adam Seth Levine, Spencer Piston and Alexander Von Hagen-Jamar argue that this has rather to do with the constitutional setup of the state: some states put higher hurdles to the amendments of their constitution. This differs from the conventional wisdom that states that jurisdictions with more liberal constitution tend to accept more liberal amendments.
The constitutional set-up and in particular the way constitutional amendments are passed have been established decades if not centuries before the debate about same-sex marriages emerged. Specifically, two aspects matter: whether citizens can place constitutional amendments on the ballots without legislative intervention, and if not, whether simple majorities from legislature and voters are sufficient. This explains better constitutional outcomes than voter attitudes.
So, institutions matter. However, the above analysis does not take into account that same-sex marriages amendments are currently evolving and have not yet reached a steady state. The picture could be very different in a decade. Whether institutions matter in the long run remains to be seen.