Access to Veteran-Centered Community Care Act - Directs the Secretary of Veterans Affairs (VA) to: (1) determine that a VA facility is not capable of furnishing the care or services required if a veteran has been waiting for 14 days or longer for an appointment to receive hospital care or medical services, including primary care, in such a facility; and (2) ensure that such care or services are provided to such a veteran at a non-VA facility under contract with the VA.

Requires: (1) any referral of a veteran who, in the course of receiving hospital care or medical services in a non-VA facility, is referred to another non-VA facility for specialty care, to be submitted to the Secretary; and (2) the Secretary, if he or she does not reject such referral during the 72-hour period following its submittal, to ensure that such specialty care is provided to the veteran at a non-VA facility under contract with the VA.