3D point clouds directly collected from objects through sensors are often incomplete due to self-occlusion. Conventional methods for completing these partial point clouds rely on manually organized training sets and are usually limited to object categories seen during training. In this work, we propose a test-time framework for completing partial point clouds across unseen categories without any requirement for training. Leveraging point rendering via Gaussian Splatting, we develop techniques of Partial Gaussian Initialization, Zero-shot Fractal Completion, and Point Cloud Extraction that utilize priors from pre-trained 2D diffusion models to infer missing regions and extract uniform completed point clouds. Experimental results on both synthetic and real-world scanned point clouds demonstrate that our approach outperforms existing methods in completing a variety of objects. Our project page is at \url{https://tianxinhuang.github.io/projects/ComPC/}.