Quiet's fate needs to e decided by you, regardless of whether you have deprived her of her entire health bar or the endurance bar. After you defeat her, Quiet should appear in a different spot on the map. Get there and approach her. If you want to shoot Quiet, you need to press the attack button,, at the moment at which your character aims at her. This is not a good idea because you will lose, this way, your opportunity to get a new buddy.
If you want to capture Quiet alive, do no do anything and just wait until your character picks her up from the ground. Then, you need to wait for the helicopter to appear in the ruins and carry her over into the vehicle. In this case, the mission ends in several cutscenes, where you do not need to perform any additional actions. After you reach the Mother Base, Quiet will be arrested.