Medicare Cost-Sharing and Welfare Extension Act of 2005

Medicare Cost-Sharing and Welfare Extension Act of 2005 - Amends title XIX (Medicaid) of the Social Security Act (SSA) to extend through September 2006 the program of cost-sharing assistance to qualifying individuals for the Medicare Part B premium (QI program).

Sets the total amount available for allocation for the QI program as follows: (1) for the period that begins on October 1, 2005, and ends on December 31, 2005, at $100 million; and (2) for the period that begins on January 1, 2006, and ends on September 30, 2006, at $300 million.

Extends through March 31, 2006: (1) the Temporary Assistance for Needy Families Program (TANF) under SSA title IV part A; (2) the separate program for abstinence education; (3) additional grants for Puerto Rico, the Virgin Islands, Guam, and American Samoa; (4) supplemental grants for population increases in certain states; and (5) the Transitional Medical Assistance program under Medicaid. Makes appropriations.

Extends the National Random Sample Study of Child Welfare and Child Welfare Waiver Authority through March 31, 2006.

Excludes drugs used to treat sexual or erectile dysfunction from coverage as prescription drugs under SSA title XVIII (Medicare), beginning in 2007, except when used to treat another condition, subject to Food and Drug Administration approval. Authorizes states to restrict or exclude such drugs from coverage, with the same exception, under Medicaid.