LEAD: Dr. Richard L. Day, retired professor of pediatrics at the Mount Sinai School of Medicine in Manhattan, died yesterday at his home in Westbrook, Conn. He was 84 years old.

Dr. Richard L. Day, retired professor of pediatrics at the Mount Sinai School of Medicine in Manhattan, died yesterday at his home in Westbrook, Conn. He was 84 years old.

Dr. Day taught at Mount Sinai from 1968 until he retired in 1971. Earlier he had been national medical director of Planned Parenthood (1965-1968); professor and chairman of the department of pediatrics at the University of Pittsburgh (1956 to 1965); professor and chairman of the department of pediatrics in the Downstate Medical School in Brooklyn (1953 to 1956) and associate professor at the College of Physicians and Surgeons, Columbia University (1935 to 1953).

He was recognized for his studies of the mechanism controlling the body temperature of premature babies and for his discovery that a yellow pigment that causes jaundice can on occasion damage a newborn baby's brain.

In 1986 he received the American Pediatric Society's highest honor, the Howland Award, for being ''the quintessential skeptical inquirer.''

Dr. Day was born in Manhattan and graduated from Harvard College in 1927 and Harvard Medical School in 1931.

He is survived by his wife, the former Ida Holt; three children, Sally Ranti of Old Saybrook, Conn.; Betsy Darlington of Ithaca, N.Y., and Kate Beare of London, and six grandchildren.