Coulter was born on April 27, 1891 in San Antonio, Texas. He graduated from West Texas Military Academy in 1911, and in 1912 obtained a commission as a Second Lieutenant of Cavalry. He initially served with the 14th Cavalry Regiment in Texas until 1916, including participation in the Pancho Villa Expedition.

During World War I he served in France as aide-de-camp to General William A. Mann, the commander of the 42nd (Rainbow) Infantry Division. After returning to the U.S. for five months as he then went back to France as commander of 2nd Battalion 508th Pioneer Infantry, an African-American unit. After some training and various positions in the U.S. he was assigned to the General Staff's Military Intelligence Division as a specialist in Latin American affairs.

He served as commander of the 4th Cavalry Regiment (1938-1940); commander of the 3rd Cavalry Brigade (1941). After the attack on Pearl Harbor, his soldiers patrolled the Mexican border. In early 1942 Coulter was assigned to command 2nd Cavalry Division.

Coulter assumed command of the 85th Infantry Division in 1943. After stateside training, he led his division in North Africa and Italy. The 85th Division fought through the Gustav and Gothic Lines during the Rome-Arno, North Apennines and Po Valley campaigns, and Coulter earned a reputation as an expert in military mountaineering and alpine warfare.

In 1948 he went to Korea as commander of the 7th Infantry Division. In 1949 he was appointed deputy commander of U.S. forces in Korea, and then commanded I Corps until its deactivation in 1950. Coulter was then assigned as deputy commander of Fifth Army, headquartered in Chicago.

After the June, 1950 invasion of South Korea, Coulter was assigned to command I Corps, reactivated as part of the Eighth Army. As the commander of Task Force Jackson, an ad hoc force of South Korean and U.S. troops, Coulter was credited with a key role in halting North Korea's advance. In September, 1950 Coulter assumed command of IX Corps, and led his organization as the supporting effort to I Corps in the U.S. counterattack against North Korea.

In 1951 Coulter was promoted to Lieutenant General as deputy commander of the Eighth Army, and was Eighth Army commander General Matthew Ridgway's liaison to the South Korean Army and South Korean President Syngman Rhee. General Coulter retired from the Army in 1952.

In 1959, Syngman Rhee, still the President of South Korea, erected a statue of Coulter to recognize his efforts to rebuild South Korea. The statue originally stood in the Itaewon District of Seoul. It was rededicated in 1977, and now stands at Seoul's San 18, Neung-dong, Kwangjin-gu.

In the 1960s, Coulter was President of the Korean Cultural and Freedom Foundation, an organization formed to recognize Korean War veterans and foster cultural exchanges between the U.S. and South Korea.

Coulter died in Washington, D.C. on March 6, 1983 and was buried at Arlington National Cemetery.

His awards and decorations included the Distinguished Service Medal (3), Silver Star (2), Bronze Star, Distinguished Flying Cross, and Air Medal (6).