Existing video stabilization methods based on 2D warping or local 3D cues often struggle to preserve full-frame content and geometric consistency, particularly under large camera motions or complex scene geometry. We present StabiGS, a novel approach that formulates video stabilization as a rendering-aware view synthesis problem. Our method reconstructs a global 3D scene representation and jointly optimizes the camera smoothness and the rendering quality to compute the stabilized trajectory. Thus, unlike filtering-based approaches, StabiGS produces stable videos while maintaining geometric consistency. To enable a comprehensive evaluation, we present a synthetic benchmark with controllable camera shakiness and accurate ground-truth poses, enabling a reliable assessment of stabilization and rendering quality. Extensive experiments and a user study demonstrate that StabiGS achieves state-of-the-art performance in stabilization and rendering quality.