Middleton was born in Beaufort, South Carolina, on the 16th of October 1872. He began his time in the US Navy in 1896 as an Assistant Surgeon after completing medical school at Columbia College in Washington DC. He served in the Naval Laboratory in New York and the USS Porter, before reporting to the USS Texas from which he served during the Spanish-American War. After the war, Middleton was promoted to Lieutenant (Junior Grade) and transferred to South Carolina’s Port Royal Naval Station. By mid-1911, he had been promoted to Lieutenant Commander and was serving on the USS Utah before transferring to the USS Florida. Both of these vessels deployed in support of the Mexican Campaign and Middleton was on board the Florida as a Surgeon when he displayed actions that would earn him the Medal of Honor. The citation reads:

For distinguished conduct in battle, engagements of Vera Cruz, 21-22 April 1914. Surg. Elliott was eminent and conspicuous in the efficient establishment and operation of the base hospital, and in his cool judgment and courage in supervising first-aid stations on the firing line and removing the wounded.

After the action at Vera Cruz, Middleton reported to the Navy Recruiting Station in New York City for recruiting duty. When the US entered World War 1, he had reached the rank of Commander and was in charge of the Naval Hospital and Supply Depot in the Philippines. After the war, he assumed command of the Naval Hospital in Washington DC and was promoted to Captain and Medical Director. Middleton retired at the rank of Rear Admiral in November of 1936 after serving as the president of the Naval Retirement Board and the Inspector of Medical Department Activities for four districts. In February of 1942, he was briefly recalled to active duty for World War 2 and was promoted to Vice Admiral before being placed back on the retired list after forty years of service. He and his wife, Alice, had one daughter and Middleton Stuart Elliott died on the 29th of October 1952, almost two weeks after his 80th birthday. He is buried with his wife in Fort Rosecrans National Cemetery: Section P, Grave 2628.