Event cameras provide high temporal resolution, high dynamic range, and low latency, offering significant advantages over conventional frame-based cameras. In this work, we introduce an uncertainty-aware refinement network called URNet for event-based stereo depth estimation. Our approach features a local-global refinement module that effectively captures fine-grained local details and long-range global context. Additionally, we introduce a Kullback-Leibler (KL) divergence-based uncertainty modeling method to enhance prediction reliability. Extensive experiments on the DSEC dataset demonstrate that URNet consistently outperforms state-of-the-art (SOTA) methods in both qualitative and quantitative evaluations.