National Trauma Center Stabilization Act of 2007 - Amends the Public Health Service Act to revise the grant program for trauma care centers by: (1) establishing three programs to award grants to public, non-profit, Indian Health Service, Indian tribal, and urban Indian trauma centers to assist in defraying substantial uncompensated care costs, further the core mission of such trauma centers, and provide emergency relief to trauma centers at risk of closing; and (2) setting forth minimum qualifications for trauma centers, including requirements for the treatment of charity or self-pay patients.