Textile Negotiator Promotion Act - Amends the Trade Act of 1974 to redesignate the Special Textile Negotiator of the United States Trade Representative (USTR) as the Chief Textiles Negotiator who shall be appointed by the President, by and with the advice and consent of the Senate. Makes it a function of the Chief Textiles Negotiator to conduct trade negotiations and enforce trade agreements regarding U.S. textile and apparel products and services.