There's an algorithm that does it. I wrote a hobby implementation ten years ago. Basically, it's a modification of the sweeping line algorithm that sweeps the sphere from pole to pole.

edit: Found the code. Looks like I instead ended up simply building a convex hull (which is the Delaunay triangulation) and deriving the Voronoi diagram from it.

For spherical geometry you should not need to do anything special, no ?

Compute the Voronoi tessellation as usual and then project the linear separators back to the sphere (by connecting points on it to centre and extending the ray to the spherical surface).