The application potential of robotic fish embedded with intelligent visual navigation algorithms in underwater autonomous operation is on full display recently. However, the existing visual navigation methods are limited by the underwater visual conditions and the motion characteristics of robotic fish. To this end, this paper proposes a novel autonomous navigation framework integrated with visual stabilization control. In practice, a stereo vision-based navigation network is proposed to generate the guidance law. On this basis, a biomimetic robotic hammerhead shark with a controllable cephalofoil is developed, and a nonlinear model predictive controller for cephalofoil stabilization relying on the dynamic model is elaborately designed. Extensive simulations and underwater experiments are conducted to validate the effectiveness and superiority of the proposed methods, which significantly enhance exploration efficiency and reduce image jitter by 26.02% compared to the traditional methods. The obtained results provide a new idea for underwater robots to autonomously explore the ocean.