In 1983, Healy received the Laurence Olivier Award for Best Supporting Actor for his role in Guys and Dolls during the 1982 theatre season. His performance of "Nicely Nicely Johnson" was praised as "show-stopping" as he sang "Sit Down You're Rockin' the Boat". He performed a mid-show encore each night. In 1986 he played the character of Buddy Plumber in a London run of the Stephen Sondheim musical Follies.
Healy died following a heart operation on October 25, 1995 in London, England.
David married Peggy Walsh and had two sons, William and Tim. He was a devoted amateur polo player and his wife was the manager of Ham Polo Club in London. Both of his sons remain polo players and the David Healy Trophy is still played for in his memory.