Health Centers Consolidation Act of 1996 - Amends the Public Health Service Act to revise provisions concerning health centers (currently, community health centers), definitions relating to such centers, and the services provided by such centers. Authorizes the Secretary of Health and Human Services to make planning, operating, and infant mortality grants for health centers that provide required primary health services and additional health services to medically underserved and special medically underserved populations (migrant and seasonal agricultural workers, the homeless, and public housing residents). Directs the Secretary to give priority in making grants to health centers in which there is a substantial incidence of infant mortality or a significant increase in the incidence of infant mortality. Specifies that grant amounts for any fiscal year may not exceed: (1) total State, local, and other operational funding for a center; and (2) fees, premiums, and third-party reimbursements received for operations by the center in such fiscal year. Authorizes appropriations.

Requires the Office of Rural Health Policy of the Health Resources and Services Administration to administer the rural health services outreach demonstration grant program. Authorizes appropriations.