Born in Bridgeport, Conn., Welch earned his bachelor's degree from the University of Pittsburgh in 1940. He received his medical degree from Yale University in 1943. Welch did postgraduate training at the Montreal Neurological Institute and at McGill University, where he received a master's in science degree in 1947. After two years of active duty in the U.S. Army Medical Corps, Welch joined the University of Colorado. During his 16 years there, Welch made great strides in cerebrospinal fluid research. He developed a mathematical model for predicting the distribution of material in the fluids of the central nervous system, and published widely on this topic.
Welch came to Harvard Medical School in 1971. He served as neurosurgeon-in-chief at Brigham and Women's and Children's Hospital until his retirement in 1987.
Welch leaves his wife Elizabeth (McRae) of Waban; five daughters, Elizabeth Buchanan and Mary Buchanan of England, Anne and Joan of Cambridge, and Stephanie Edwards of Barrington, R.I.; and six grandchildren.