Prescott Marine Mammal Stranding Program Amendments of 2004 - Amends the Marine Mammal Protection Act of 1972 to authorize appropriations for FY 2005 through 2010 for: (1) the John H. Prescott Marine Mammal Rescue Assistance Grant Program to provide grants to eligible stranding network participants for the recovery or treatment of marine mammals, the collection of data from living or dead stranded marine mammals for scientific research regarding marine mammal health, and related facility operation costs; and (2) the Marine Mammal Unusual Mortality Event Fund to compensate persons for certain special costs associated with the contingency plan for responding to any unusual mortality event, for reimbursing any stranding network participant for costs incurred in preparing and transporting tissues collected with respect to such event for the Tissue Bank, and for care and maintenance of a marine mammal seized for lack of a permit.