First Lieutenant Greer was a member of Company I, 3rd Battalion, 7th Infantry Regiment, 3rd Infantry Division. He was seriously Wounded in Action by the enemy in South Korea on May 23, 1951 and returned to duty on July 15, 1951. He was Killed in Action while fighting the enemy in North Korea on September 11, 1951. For his leadership and valor, First Lieutenant Greer was awarded the Silver Star, the Purple Heart with Oak Leaf Cluster, the Combat Infantryman's Badge, the Korean Service Medal, the United Nations Service Medal, the National Defense Service Medal and the Korean War Service Medal.

Lieutenant Greer is buried with his parents, Thomas William Greer, Lieutenant Colonel, United States Army, in Section 3 of Arlington National Cemetery.