Constantine was born in New York City into the house of Sidamon-Eristavi, claiming descent from the medieval kings of Aragvi. He was the son of Prince Simon Sidamon-Eristoff, a Georgian military officer, who emigrated to the United States after the Bolsheviks invaded Georgia in 1921, and Anne Tracy, a descendant of John Bigelow, an American diplomat in the mid-19th century. In 1957 he married Anne Phipps Sidamon-Eristoff, a granddaughter of Andrew Carnegie's partner, Henry Phipps.