Dr. Godding was born in Wichenden, Massachusetts. Graduating from Dartmouth College (1854) with an A.B. degree, he studied medicine with his physician father and took lectures at the College of Physicians & Surgeons in New York, earning an M.D. at the Medical College at Castleton, Vermont. He practiced with his father for 18 months until he was appointed assistant physician at the State Hospital for the Insane in Concord, New Hampshire. In 1862, he resigned his position and entered private practice in Fitchburg, MA.
In 1863, he served at the Government Hospital for the Insane in Washington, D.C. (now St. Elizabeths Hospital) as a second assistant physician and remained there for seven years. In 1870, he was appointed Superintendent of the State Hospital for the Insane in Taunton, MA, and in 1877, he returned to St. Elizabeths Hospital in Washington, D.C., as Superintendent, where he remained until his death.
Dr. Godding made many improvements at St. Elizabeths, including increasing the number of buildings, improving the grounds and accommodations for patients, and adding land for farming purposes. In addition to his contributions to the psychiatric literature, with 20 articles noted (1882–96), Dr. Godding was President of the Association of Medical Superintendents of American Institutions for the Insane (1889–1890).