Youth Drinking Elimination Act - Amends the Public Health Service Act to direct the Secretary of Health and Human Services to make grants to enable public and nonprofit private entities, Indian tribes, and tribal organizations to develop plans for and to carry out school (including college) and community based programs for the prevention of alcoholic beverage consumption by persons who have not attained the legal drinking age. Requires a prospective grantee to provide assurances that it will: (1) use science based and age appropriate approaches; and (2) involve local public health officials and community prevention program staff in the planning and implementation of the program. Authorizes appropriations.