Morgan was born in San Gabriel, California, and was taught to swim in the ocean at Huntington Beach by his parents. He joined the local swim team and worked as a junior lifeguard. The family moved to Argentina, his father's home country, when Morgan was 13. Morgan was finding success as a competitive swimmer and he completed a 20K marathon river swim at the age of 15. While exercising, an accident with a springed exerciser he was holding left Morgan with two detached retinas. Over the next two years his eyesight deteriorated, until in 1980 at age 18 he lost all vision. In 1984, while studying mathematics at University of California, Irvine, he was selected for the All-American Team for open water swimming.
Morgan was competing as a blind swimmer in B1 category events when he attended the International Games for the Disabled, subsequently known as the Paralympic Games, in 1984. He found great success, finishing the tournament with five gold medals. In 1988 he attended the University of California, Berkeley to do his masters degree and consequently missed the Paralympic Games that year. After completing his studies in 1991, he took part in a 35-mile (56 km) open water marathon in Argentina which he completed in 12th position with a time of 8 hours and 24 minutes. Morgan returned for the 1992 Games in Barcelona, again finding success by winning eight golds in backstroke, butterfly, freestyle, and medley events, as well as silver in two breaststroke distances.
Morgan was inducted into the United States Olympic Hall of Fame in 2008 and the United States Association of Blind Athletes' Hall of Fame in 2011.
Aside from swimming, Morgan has pushed himself in other athletic endeavours, including tandem cycling and the successful ascent of Mount Kilimanjaro.