Office of Attending Physician Independence Act This bill addresses the appointment and oversight of the Attending Physician (a medical officer who is currently assigned by the the Navy to provide emergency medical assistance to Members of Congress, Justices of the Supreme Court, congressional staff, and visitors to the Capitol). The bill establishes a commission to recommend individuals to the President for appointment to the position of Attending Physician. The commission shall recommend at least three individuals who are medical officers of the Navy for appointment. The President shall appoint the Attending Physician from the individuals recommended by the commission with the advice and consent of the Senate for a term of 10 years. The Office of Attending Physician shall be subject to oversight by the Committee on House Administration and the Senate Committee on Rules and Administration.