Woodward graduated from MIT in June, 1980, with a degree in Physics. He attended graduate school at the University of Texas at Austin, working in the Center for Relativity and then the Fusion Research Center. His thesis research involved using optical spectroscopy to investigate neoclassical plasma rotation in the Texas Experimental Tokamak fusion reactor. He received his Masterís degree in 1988 and joined the US Navy, reporting to Aviation Officer Candidate School in Pensacola, Florida. He was commissioned in January 1989, and earned his wings as a Naval Flight Officer in March 1990. He reported to NAS Whidbey Island, Washington, where he completed initial Bombardier/Navigator training in the A-6E Intruder and was then assigned to the Green Lizards of Attack Squadron 95. Woodward made two deployments with VA-95 aboard the USS Abraham Lincoln (CVN-72) in support of Operations Desert Storm (post-cease fire), Southern Watch, and Somalia. He was then assigned to the Naval Strike Warfare Center in Fallon, Nevada, where he served as Weaponeering Officer and Contingency Cell Officer. In 1995, he was selected to attend the U.S. Naval Test Pilot School at Naval Air Station Patuxent River, Maryland, and graduated with distinction in July 1996. Upon graduation, he was assigned to the Air Vehicle/Stores Compatibility Department at the Naval Strike Aircraft Test Squadron in Patuxent River. While at NSATS, he cross-trained in the F/A-18 Hornet and tested stores and systems for the F/A-18B, F/A-18D, and F/A-18F aircraft. Woodward was assigned to the Strike Aircraft Test Squadron when he was selected for the astronaut program.
Woodward has logged over 1,500 flight hours in more than 25 different aircraft and has 265 arrested landings.