Bilateral control is one of the methods to realize a teleoperation with force feedback. However, communication time delay seriously deteriorates operationality and stability of bilateral control. Although 3-channel bilateral control has been proposed to improve operationality, the system becomes unstable in contact motion. Damping injection is an effective method to stabilize the contact motion by producing viscous force. However, this method produces large operational force and deteriorates operationality. Many researchers tried to inject damping without deteriorating operationality. However, their methods are affected by several situations/factors such as delay time or operator's motion speed. As a result, their methods require a redesigning procedure in order to deal with such situations/factors. In this paper, automatically-tuned damping injection is proposed to stabilize the contact motion in 3-channel bilateral control. In the proposed method, damping gain is automatically determined by monitoring the Lyapunov stability of the force control. The effectiveness of the proposed method is verified by experiments.