Veterans Dialysis Pilot Program Review Act of 2013 - Prohibits the Secretary of Veterans Affairs (VA) from expanding the dialysis pilot program (established by the Secretary in 2009 to provide dialysis care to patients at certain VA outpatient facilities) to any facility other than an initial facility (one of four outpatient facilities the Secretary identified to participate in the program) until after: (1) the Secretary has implemented the program at each initial facility for at least two years, (2) an independent analysis of the program has been conducted at each initial facility, and (3) a report from the Secretary to Congress (required under this Act) on such analysis has been submitted.