I argued in my 2014 book, White Voters in 21st Century America, that the GOP can partially offset its demographic problem by boosting its share of the non-Hispanic white vote. There is a temporal limit to this, of course, as eventually even massive majorities among white voters will not be enough to win national elections without at least some support for non-whites. But in the short-term, it may be easier for the GOP to get a few million additional white votes than a similar number of minority votes. A major focus on gay marriage, however, would probably have hindered such an effort.
Public opinion on gay marriage has shifted dramatically over the last decade, to the point where a majority of the electorate supports gay marriage. In fact, whites are more supportive of gay marriage than non-whites (57 percent of whites now support marriage equality). There was some talk during the Bush years about how the GOP's more conservative stance on gay marriage and related social issues would help the party make inroads among minority voters, as these voters were, on average, somewhat more traditionalist in their views on these subjects. Well, the results are in, and we now know that being anti-gay marriage did not cause large numbers of African Americans and Latinos to vote Republican.
It is difficult to say whether gay marriage was an issue that determined vote choice for most people, but if boosting its support among whites is a goal for the GOP, it makes little electoral sense to take a hard position on gay marriage when a majority of whites disagree with the traditional Republican stance. I argued the following in White Voters:
[I]t is not inconceivable that many whites are turned off by the Republican Party because of the party's apparent hostility to gay and lesbian Americans ... The arrival of a post-gay marriage era in American politics, should it ever arrive, may actually be beneficial to the Republican Party's standing among whites, particularly when we consider the many other issues that apparently unite large numbers of whites are issues where whites tend to be in agreement with the GOP's platform. [p. 13]
Although it was clearly not the intention, the Obergefell decision may have been a gift to the GOP. Had the Supreme Court not taken the issue off of the table, a large and vocal minority within the Republican Party would have insisted that the party maintain its opposition to gay marriage. Because the issue has been removed from the political arena, GOP leaders have a reasonable cover to quietly drop the subject and move on to other things. The anti-gay marriage evangelicals are almost certainly going to remain Republican in any event, and there is now one fewer issue where the GOP platform is at odds with public opinion.