Metzger was born to a Jewish family in Mainz on March 23, 1932, to a hardware store owner and a homemaker.
At the suggestion of his sisters, Metzger's father moved to the United States in 1937, followed by his wife and sons in January 1938.
Henry Metzger attended the Bronx High School of Science, as did his brother, then earned a bachelor's degree from the University of Rochester in 1953, followed by a medical degree at the Columbia University College of Physicians and Surgeons in 1957.
After two years of post-doctoral study with Seymour Jonathan Singer, funded partly by the Helen Hay Whitney Fellowship, Metzger returned to the National Institutes of Health, working primarily in the National Institute of Arthritis and Musculoskeletal and Skin Diseases.
Six years later, Metzger received the American Association of Immunologists Lifetime Achievement Award, the AAI's highest honor.