Apparently, that would violate the State Constitution.
Mon Feb 11, 2019 2:30pm (XFF:

“One could make an argument that allowing Smith [appointed rather than elected] to become Governor would defeat the intention of Article V, Section 16. That section lays out in great detail succession to the office of Governor,” he [ A.E. Dick Howard, Warner-Booker distinguished professor of international law at the University of Virginia and executive director of Virginia’s Commission on Constitutional Revision ] wrote.

“To allow the process of resignation and the filling of vacancies to result in the installation of a Governor who lacked any mandate of the people would be inconsistent with Section 7’s purpose and structure. I can imagine the argument, therefore, that the careful and specific process laid out in Section 16 would control over the general provision in Section 7.”

Howard helped draft the current constitution, so he’s probably more familiar with its sections than you are. What he’s saying, in essence, is that the elevation of someone to the office who has no mandate from the state’s voters would violate the spirit of the articulated succession system. That succession system might be determined (by a court, presumably) to hold more weight than Northam’s handpicked lieutenant governor, and Herring would, again, be appointed governor."

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