Rural Veterans Health Care Act of 2006 - Directs the Secretary of Veterans Affairs to: (1) conduct an outreach program to identify and provide information to veterans who served in Operations Iraqi Freedom or Enduring Freedom and who reside in rural communities in order to enroll those veterans in the Department of Veterans Affairs (VA) health care system; (2) expand access to Vet Centers in rural areas; (3) develop and implement a plan for improving veterans access to health care in rural areas; (4) establish a health information technology pilot program to ensure quality care for rural veterans through VA facilities; (5) establish the Advisory Committee on Rural Veterans; (6) designate at least four VA health care facilities as centers of rural health research, education, and clinical activities; (7) include rural veterans in a current VA grant program for homeless veterans with special needs; (8) enhance the education, training, retention, and recruitment of health professionals in rural areas; and (9) expand VA health care presence in Native American, Native Hawaiian, and Native Alaskan rural communities.