From Justice John Paul Stevens, writing on April 11, 2014 in the Washington Post, commenting on Justice Antonin Scalia's ruling in District of Columbia v. Heller: 

"Prohibitions on carrying concealed weapons, or on the possession of firearms by felons and the mentally ill, and laws forbidding the carrying of firearms in sensitive places such as schools and government buildings or imposing conditions and qualifications on the commercial sale of arms are specifically identified as permissible regulations."

States (or commenwealths) are free from federal governmental interference on this matter. If Virginia wants to ban weapons from public buildings, it is free to do so. And I think we should. Other than the military and the police, I don't see the need for weapons in public buildings in Virginia.