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. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the problem of vision-based underwater autonomous navigation for a biomimetic robotic fish that possesses underwater visual stability and good maneuverability. The existing visual navigation networks usually generate unexpected navigation instructions when dealing with complex or ambiguous underwater scenes. Additionally, image jitter caused by the rhythmic motion of robotic fish can lead to navigation failure. This paper suggests an integrated navigation framework that includes a biomimetic platform design, a visual stabilization controller, and an intelligent underwater navigation network. Specifically, a novel sphyrnidae-inspired robotic shark is designed as a new platform with superior motion performance. To enhance underwater visual stability, a nonlinear model predictive control-based visual stabilization controller is proposed. Furthermore, a deep stereo attention navigation network based on a parallax attention mechanism is proposed to improve the generalization of vision-based autonomous navigation. A series of underwater search experiments on the robotic shark demonstrate the effectiveness and superiority of the proposed navigation framework. Hopefully, our proposed methods can provide valuable guidance and support for universal underwater robot navigation to accomplish practical marine tasks, such as underwater rescue, resource exploitation, biological observation, and so on.