No, today is not the day on which seamstress Betsy Ross delivered the first American flag, although that makes a good story.
However, the Second Continental Congress did adopt the first American flag by resolution dated June 14, 1777:
Resolved, that the flag of the United States be thirteen stripes, alternate red and white; that the union be thirteen stars, white in a blue field representing a new constellation.Our crack research staff found this site which claims that the first celebration of Flag Day can be traced to 1885 when "BJ Cigrand, a schoolteacher, arranged for the pupils in the Fredonia, Wisconsin Public School, District 6, to observe June 14... as 'Flag Birthday.'" Cigrand, who was later a resident of Batavia, Illinois, became a tireless proponent of the day and is sometimes described as the Father of Flag Day. In 1911, a major national fraternal organization, the Benevolent and Protective Order of Elks, made Flag Day celebrations mandatory in all of its lodges. President Woodrow Wilson issued a proclamation that officially established June 14 as Flag Day in 1916. In 1949, the observance of Flag Day was established by Act of Congress (see, 36 U.S.C. § 110).
If you need something else to celebrate, the U.S. Army was born on this date in 1775. Hooah and happy birthday.