Answer: He was 14.

According to the Doogie Howser M.D. Wikipedia page (and, if you remember, the opening credits), “Howser got a perfect score on the SAT at the age of six. He completed high school in nine weeks, graduated from Princeton University in 1983[3] at age 10, and finished medical school four years later. At age 14, Howser was the youngest licensed doctor in the country.”