Stockton was born in New Castle Hundred, New Castle County, Delaware, on April 1, 1781, son of John and Nancy Griffin Stockton. His father was a brigadier general in the War of 1812, leading the 1st Brigade of Delaware militia. He married Fidelia Johns, daughter of Chancellor Kensey Johns, and they had five children, Thomas, William, James, Elizabeth, and Fidelia. They lived in the Kensey Johns-Van Dyke house at 300 Delaware Street and were members of the Immanuel Episcopal Church in New Castle.
After graduating from Princeton, Stockton joined the Delaware militia. During the War of 1812 he was part of the attack on Fort George on the Niagara River and fought at Lundy's Lane. He was promoted to a major in the 42nd Infantry, and he retired from the U.S. Army in 1825.