We present a novel region growing algorithm for plane extraction of organized point clouds using the spherical convex hull. Instead of explicit plane parameterization, our approach interprets potential underlying planes as a series of geometric constraints on the sphere that are refined during region growing. Unlike existing schemes relying on downsampling for sequential execution in real time, our approach enables pixelwise plane extraction that is highly parallelizable. We further test the proposed approach with a fully parallel implementation on a GPU. Evaluation based on public data sets has shown state-of-the-art extraction accuracy and superior speed compared to existing approaches, while guaranteeing real-time processing at full input resolution of a typical RGB-D camera.