Henry Wharton Conway was born on 18 March 1793 near Greeneville, Tennessee in Greene County, Tennessee. Conway was educated by private tutors. He was commissioned as an Ensign in the United States Navy during the War of 1812 and was promoted to Lieutenant in 1813.
In 1817 became a clerk in the U.S. Treasury and served in that position until he moved to the Missouri Territory in 1818. In 1820 he moved to the Arkansas Territory.
In Arkansas he was appointed as receiver of public moneys and served in that position in 1820 and 1821. He was elected as a delegate to the Eighteenth, Nineteenth, and Twentieth Congresses.
Conway was mortally wounded on 29 October 1827 during a duel with Robert Crittenden who was Secretary of the Arkansas Territory. Conway lingered for a couple of weeks after the duel but succumbed to his wounds.
Henry Wharton Conway died on 9 November 1827. He is buried in the Arkansas Post cemetery.
Conway was the brother of Governor James Sevier Conway and Governor Elias Nelson Conway. He was first cousin to Senator Ambrose Hundley Sevier and Governor Henry Massey Rector. He was also third cousin to Confederate General James Lawson Kemper.
Conway County, Arkansas is named for him.