Dr. Moore graduated from medical school at The University of California, San Francisco. He then entered general surgery training at that institution and completed training in 1964. He spent two years in the United States Army as chief of  surgery at the 98th general hospital in Germany. Upon completion of his military obligation, He returned to San Francisco to begin specialty training in vascular surgery. Upon completion of his vascular fellowship, he was appointed as assistant professor of surgery at the University of California, San Francisco and Chief of Vascular Surgery at the Veterans Administration Medical center. In 1977, he was recruited by the University of Arizona to start a vascular surgery program at that institution and was appointed Professor of Surgery, Chief Section of Vascular Surgery at the University of Arizona. In 1980, he was recruited by UCLA to return to California and head the vascular surgery program at that institution where he has remained until the present.

Dr. Moore developed an early interest in cerebrovascular disease and is a recognized expert in the surgical management of carotid artery disease as a strategy for stroke prevention. He has been a participant and leader in the major clinical trials including the VA asymptomatic carotid stenosis study, The North American Symptomatic Carotid Stenosis Study (NASCET), The Asymptomatic Carotid Stenosis Study (ACAS), and most recently, the Carotid Revascularization Endarterectomy vs. Stenting Trial (CREST). He has edited the definitive text book entitled Surgery for Cerebrovascular Disease.

More recently, Dr. Moore developed an interest and expertise in the less invasive approach to treating thoracic and abdominal aortic aneurysm using the stent-graft approach.. In 1992 he became involved in the investigation of stent-graft repair of abdominal aortic aneurysm and on February 10th, 1993 implanted the first commercially manufactured stent graft, Endovascular Technologies (EVT), worldwide. Since then, he led the UCLA program to become a leader in aortic stent-graft clinical experience.