Murray was born to william A. and Mary (née DePasquale) Murray, and grew up in Milford, Massachusetts. He was of Irish and Italian descent. A star athlete at the Milford High School, he excelled in football, ice hockey, and baseball. Upon graduation, Murray attended the College of the Holy Cross intending to play baseball; however, baseball practices and lab schedules conflicted; forcing him to give up baseball. He studied philosophy and English, earning a degree in humanities in 1940. Murray later attended Harvard Medical School. After graduating with his medical degree, Murray began his internship at the Peter Bent Brigham Hospital. During that time, he was inducted into the Medical Corps of the U.S. Army.