The problem of stabilizing steady gliding is critical for an underwater glider, which is subject to many non-negligible disturbances from the aquatic environment. In this paper, we propose a new systematic controller design and implementation approach for the stabilization problem, including a nonlinear, passivity-based controller and a nonlinear model-based observer, where the actuation is realized through a whale tail-like control surface. The controller is designed based on an approximation of a reduced model that is obtained through singular perturbation analysis, and consequently, it does not require full state feedback. The local stability of the full closed-loop system is established through linearization analysis. The nonlinear observer is designed to estimate the velocity-related system states, which are difficult to measure for such low-speed underwater vehicles. Simulation results are first provided to demonstrate that the proposed controller achieves rapid convergence in stabilization and the proposed observer has good performance especially in robustness against measurement noise. Experimental results using a gliding robotic fish are presented to support the effectiveness of both the controller and the observer.