He was born at Princeton, New Jersey into a political family; his father Richard Stockton was a U.S. Senator and Representative, and his grandfather, another Richard Stockton, signed the Declaration of Independence.

Robert was appointed a midshipman in the U.S. Navy at the age of 16, serving at sea and ashore during the War of 1812. After that conflict, Lieutenant Stockton was assigned to ships operating in the Mediterranean, in the Caribbean and off the coast of West Africa. While on the latter station, he helped negotiate a treaty that led to the founding of the state of Liberia. During the later 1820s and into the 1830s, he primarily devoted his attention to business affairs in New Jersey. The birth of his son John P. Stockton, later also a U.S. Senator representing New Jersey, occurred during this time.

In 1838, Stockton resumed active naval service as a captain. He served in the European area, but took leave in 1840 to undertake political work. Offered the post of U.S. Secretary of the Navy by President John Tyler in 1841, he declined the offer, but worked successfully to gain support for the construction of an advanced steam warship with a battery of very heavy guns.

This ship became USS Princeton, the Navy's first screw-propelled steamer. The ship was designed by John Ericsson. Stockton commanded her when she was completed in 1843. Although he was the deviser of a defective gun, Captain Stockton was absolved of responsibility for the February 1844 explosion of the gun, the Peacemaker, on board the ship. The explosion killed two cabinet officers and several others. After the disaster, he pointed the blame towards Ericsson, who had nothing to do with Stockton's design. Stockton, with a wholly inadequate understanding of the basic principles of heavy weapons design, had attempted to copy an Ericsson design, with these fatal results. Stockton was so successful in deflecting blame that he prevented any payments to Ericsson for his work on the project.

With the temporary title of Commodore, Stockton commanded naval forces in the eastern Pacific Ocean, and was instrumental in taking California from Mexico at the Battle of Rio San Gabriel and Battle of La Mesa. He served as the first military governor of California. There, as he had earlier quarreled with Ericsson, he quarreled with his fellow-commander, General Stephen Watts Kearny.

Stockton resigned from the Navy in May 1850 and returned to business and political pursuits. In 1851 he was elected as a Democrat from New Jersey to the United States Senate, where he sponsored a bill to abolish flogging as a Navy punishment. He resigned on January 10, 1853 to serve as president of the Delaware and Raritan Canal Company, a position he held until 1866.

In 1861 he was a delegate to the unsuccessful conference that attempted to settle the secession crisis. In 1863, he was appointed to command the New Jersey militia when the Confederate Army invaded Pennsylvania. Captain Robert F. Stockton died at Princeton.

Four U.S. Navy ships have been named USS Stockton in honor of Robert F. Stockton. The cities of Stockton, California and Fort Stockton, Texas, and Fort Stockton, San Diego, California (now a ruin, but occupied during the Mexican-American War) in are named in his honor, as is Stockton Street in San Francisco, California.