I could find no state that used George Washington's family coat of arms as a bisis for the design of its flag, but I did find that, appropriately, the flag of Washington, D.C. used the coat of arms.
The flag of Washington, D.C., was adopted in 1938. Since Washington, D.C., had no official flag, a commission was formed in 1920 to find a design. Headed by A. E. Dubois, the final design was chosen on October 15, 1938. The design was based on the shield from George Washington's family's coat of arms. - Enchanted Learning
The official flag of the District of Columbia -------- Three red stars above two red bands on a white shield are the charges on the Washington family coat of arms and the model for the American Flag.
Washington Coat of Arms as a stained glass window in the Selby Abbey, U.K.
---- Flag: Adopted in 1938. Design was based on the shield from George Washington's family coat of arms. - DC.gov ------- In a letter written near the end of his life to an English genealogist, Washington claimed that he knew little and cared little about his English ancestry. He may not have, but he had sufficient family pride to have a Washington coat of arms imprinted on his bookplate in 1771, displayed here next to a similar bookplate, of slightly earlier date, for a distant English cousin. He employed a seal bearing his family coat of arms on many of his personal letters, and he had a large wooden carving of the crest on the wall at Mount Vernon.
