Developing low-cost and easy-to-deploy table tennis training robots is a significant challenge, largely due to the stringent, time-consuming, and error-prone camera calibration required by traditional visual servoing systems. This paper directly addressed this problem by proposing a robust, calibration-free visual control framework that enables a robotic manipulator to perform training tasks using a single monocular camera with completely unknown intrinsic and extrinsic parameters. This is a depth-independent visual feedback controller that directly translates the pixel error between the racket and the ball into control actions. To handle the unknown camera projection model, a computationally efficient adaptive law was designed. This law utilizes a regression matrix to isolate the unknown parameters and updates them in real time, completely eliminating the need for any pre-calibration process. The stability of the entire closed-loop system was rigorously proven via the Lyapunov method, guaranteeing the convergence of the tracking error. Extensive simulations on a manipulator validated the method's effectiveness and practicality. The results demonstrated that the system achieves rapid and precise tracking, with the image error converging completely to zero in under 0.8 second. The controller's robustness was further confirmed in scenarios with varying target positions and continuous multi-stroke sequences, demonstrating its suitability for dynamic and realistic training environments.