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Unmanned aerial vehicles could accurately accomplish complex navigation and obstacle avoidance tasks under external control. However, enabling unmanned aerial vehicles (UAVs) to rely solely on onboard computation and sensing for real-time navigation and dynamic obstacle avoidance remains a significant challenge due to stringent latency and energy constraints. Inspired by the efficiency of biological systems, we propose a fully neuromorphic framework achieving end-to-end obstacle avoidance during navigation with an overall latency of just 2.3 milliseconds. Specifically, our bio-inspired approach enables accurate moving object detection and avoidance without requiring target recognition or trajectory computation. Additionally, we introduce the first monocular event-based pose correction dataset with over 50,000 paired and labeled event streams. We validate our system on an autonomous quadrotor using only onboard resources, demonstrating reliable navigation and avoidance of diverse obstacles moving at speeds up to 10 m/s.