Despite their difficulties in the last few election cycles, the GOP can certainly close the gender gap and bring single women to their side. There are already plenty of women who identify with conservative/Republican ideology. The key for the GOP is to figure out how to best sell themselves as a brand to female voters, and capitalize on that.
The GOP for the past two elections has been seen as out of touch with the American people. They've been made to look as if they are against minorities, they prefer men over women, and hate gay people. Even though those ideas are made up by the liberal media, it's stuck on them. If they were to appeal more to single women, they could definitely close that gender gap. Could they completely take over it? No, but they'd be making strides.
I'm looking at this as "would accomplishing that close the gender gap" and yes, it would. Minorities and women are two groups that loathe the GOP overall and they need to work on getting back on their good side. However, I'm not sure they can actually be successful in doing so. This past election in particular, the anti-women sentiment from the more vocal members of the party was ridiculous. I'd imagine women are more reluctant than ever to give them a chance.
The GOP is against abortion, equal pay, and a host of other things that actually kind of are important to that demographic. The GOP also hates single mothers, and a lot of single women are likely to have either friends or relatives that are single mothers.
The problem is that the GOP can't reverse the trend with single women and actually gain followers in that demographic without alienating their base.