Veterans Access to Care Act of 2017 This bill authorizes and appropriates funds for: (1) Department of Veterans Affairs (VA) health care, and (2) the veterans choice program. The VA shall conduct a program to increase by up to 1,500 the number of graduate medical education residency positions at covered facilities. The bill expands: (1) eligibility for the VA family caregiver program to include members of the Armed Forces or veterans who were seriously injured or became ill on active duty prior to September 11, 2001; and (2) services to caregivers of veterans under such program to include child care, financial planning, and legal services. The support program for caregivers of covered veterans is terminated on October 1, 2022, except that any caregiver activities carried out on September 30, 2022, shall be continued. The VA may carry out specified major medical facility leases in Alabama, Colorado, California, Florida, Georgia, Indiana, Maine, Michigan, Massachusetts, Montana, New York, Pennsylvania, South Carolina, Texas, and Virginia. The VA may record as a U.S. government obligation amounts owed for hospital care or medical services furnished at non-VA facilities when a health care provider's payment claim is approved, rather than when the VA authorizes such care or services.