The USVI, an unincorporated and organized territory, is governed under the provisions of the Revised Organic Act of the Virgin Islands, enacted in 1954.
To follow up, the US Constitution and its applicability in the USVI is something of a gray area. A series of SCOTUS decisions known as the Insular Cases will likely be used to defend the Governor's actions. It's not as clear-cut as the NRA case against New Orleans in 2006.