Muligan attended Fordham University as an undergraduate, receiving his A.B. in 1939, and Fordham Law School from which he graduated in 1942. After serving in the United States Army Counter Intelligence Corps during World War II, Mulligan became a law professor at Fordham in 1946. Mulligan taught at Fordham for 25 years, including as assistant dean from 1954 to 1956 and dean from 1956 to 1971. In 1961, he was named the Wilkinson Professor of Law.
In 1971, President Richard M. Nixon named Mulligan as a Judge of the United States Court of Appeals for the Second Circuit, headquartered in New York City. Mulligan served as a judge for 10 years. He left the bench in 1981, stating that the salary for federal appellate judges was too low to provide for his family.
In addition to his legal career, Mulligan was a successful public speaker delivering acclaimed humorous and serious remarks to a variety of organizations from bar associations to Irish-American civic groups. A collection of Mulligan's after-dinner speeches was edited and posthumously with an introduction by Mulligan's son, William Hughes Mulligan, Jr., under the title Mulligan's Law: The Wit and Wisdom of William Hughes Mulligan (Fordham University Press 1997).
A summer intramural moot court competition at Fordham Law, for rising second-year students, is named for Mulligan. Students who do well receive invitations to join the Fordham Moot Court Board.