Fair and Reliable Medical Justice Act - Amends the Public Health Service Act to authorize the Secretary of Health and Human Services to award up to ten demonstration grants to states for the development, implementation, and evaluation of alternatives to current tort litigation for resolving disputes over injuries allegedly caused by health care providers or health care organizations. Requires such states to: (1) develop such an alternative to current tort litigation; and (2) promote a reduction of health care errors by allowing for patient safety data related to such disputes to be collected and analyzed by organizations that engage in efforts to improve patient safety and the quality of health care.