Nope, different issues. Driver's licenses are recognized in all states because all states and Canadian provinces have signed on to Drivers License Compact, passing legislation to that effect. Marriages (of all types) are recognized, in part, under the Constitution's Full Faith and Credit Clause which applies to "public acts, records, and judicial proceedings of every other state." Driver's licenses, firearms licenses, electrician licenses, etc. are not generally considered to be "public acts, records, and judicial proceedings..."So, Virginia can also refuse to recognize driver's and marriage licenses from other states as well?
Virginia is shaping up to be an interesting 'battleground' state. You have a generally pro-2A population with the two most important constitutional offices held by anti-2A politicians. This a great example of how the concept of fundamental rights do not (and should not) track to political ideology. Although Governor McAuliffe ran on a gun control platform, I don't believe that his election hinged on this issue. Don't be surprised if either, or both, of these folks are turned out of office when Virginian's realize that their reciprocity has been extinguished.
Vermont is an interesting model here. An uber liberal state (which I sometimes call my second home) with a near perfect record on gun control. Vermonters like things they way they are. They can stay liberal because most of the politicians know not to F with people's guns.