Fleischer died Saturday of natural causes at the Motion Picture and Television Hospital in Woodland Hills, said his son, Mark.
Mark Fleischer said he remembered his father as a gentle man who always put family first.
"My parents made a great effort to insulate their children from the craziness of Hollywood," he said. "They made sure our lives were as normal as possible."
The director's father, Max Fleischer, and his uncles Dave and Louis, pioneered animated shorts in New York, starting in 1920 with the innovative "Out of the Inkwell" series. In the 1930s, they became rivals to Walt Disney with their popular Betty Boop and Popeye the Sailor comedy shorts.
A quiet-spoken but firm-minded director, Richard Fleischer never achieved the recognition of his more flamboyant contemporaries, but his name was on a wide variety of well-known films, including "Fantastic Voyage" (1966); "Doctor Dolittle" (1967); "The Boston Strangler" (1968); "Che!" (1969); "The New Centurions" (1972); "Soylent Green" (1973); "Mr. Majestyk" (1974); "Mandingo" (1975); "Conan the Destroyer" (1984) and "Red Sonja" (1985).
Gov. Arnold Schwarzenegger, who starred in "Conan the Destroyer," praised Fleischer as "a true Hollywood legend."
"He was a man of great talent and an extraordinary director who leaves behind a legacy of amazing films," Schwarzenegger said in a statement.