Laplacian operator is a basic tool for image processing. For an image with regular pixels, the Laplacian operator can be represented as a stencil in which constant weights are arranged spatially to indicate which picture cells they apply to. However, in a discrete spherical image the image pixels are irregular; thus, a stencil with constant weights is not suitable. In this paper a spherical Laplacian operator is derived from Gauss's theorem; which is suitable to images with irregular pixels. The effectiveness of the proposed discrete spherical Laplacian operator is shown by the experimental results.