Couch received a Bachelor of Arts degree from Vanderbilt University in 1952. Four years later he was given a Doctor of Medicine degree from the same university.
Couch began his career as an Intern at Vanderbilt University Hospital in 1956 and held it for a year. In 1957, he took a position of a clinical associate at the National Institute of Health, where he worked until 1959. Then in 1961, Robert was appointed a senior investigator and a head of clinical virology section in 1965 at the same institute.
A year later he held the position of an associate professor at Baylor College of Medicine. Also Couch worked as a professor of microbiology, immunology and medicine at the same college from 1971 to 2000. Since 2000, he has been a professor emeritus at Baylor College of Medicine.
Couch is a member of American College of Physicians, American Association for the Advancement of Science, Society of Experimental Biology and Medicine, American Society of Microbiology, Infectious Diseases Society of America, American Association of Immunologists, American Federation of Clinical Research, American Society of Clinical Investigation, Southern Society of Clinical Investigation, American Association of Physicians, American Society of Epidemiology and American Society of Virology.
On April 23, 1955 Robert Couch married Katherine Frances Klein. They have 4 children.