In 1891 Thompson moved to the United States and became professor and first chairman of the Department of Surgery at the University of Texas Medical Branch in Galveston, remaining in that position until his death in 1927. During thirty-six years in Galveston Thompson became one of the country's most distinguished surgeons. He was a fellow of the Royal College of Surgeons. In 1902 he was vice president of the Texas Medical Association.
He was a founding member and first vice president of the American College of Surgeons (1913), president and founder of the Texas Surgical Society (1915), president of the Southern Surgical Association (1920), and fellow and first vice president of the American Surgical Association (1922). In 1925 he received an honorary doctor of laws degree from Baylor University.
Thompson contributed more than seventy-five publications to the literature of surgery, dealing with such topics as the diagnosis and treatment of anal fistulae, hepatic cancer, appendicitis, tumors of the face and neck, cleft palate, and kidney malformations. Thompson married Eleanor Waters Roeck of Galveston on May 16, 1896, and they became the parents of four daughters and four sons. All four sons received medical degrees from UTMB.