Final answer:
The president, who is the Commander in Chief, approves the release or discharge of officers serving indefinite active duty tours by presidential appointment.
Step-by-step explanation:
The release or discharge from their appointment for officers serving indefinite active duty tours by appointment of the president is approved by the president, who holds the position of Commander in Chief of the Army and Navy of the United States. Under Article II, Section 2 of the Constitution, the president has broad powers related to the military. These powers include the authority to grant reprieves and pardons, as well as the ability to appoint and, implicitly, discharge military officers. In cases where officers are serving by presidential appointment, it is understood that the president has the authority to approve their release or discharge.
When it comes to the broader context of presidential powers and military operations, the president can take necessary measures to repel attacks against the United States and the authorization of military force is given by Congress; however, the president must seek congressional approval for troop deployments beyond 60 days. This balance between executive and legislative power is designed to ensure that military actions are subject to democratic oversight.