It is not exactly possible (as it is also not by using a texture), but of course you can parametrize the surface of a sphere and compute the parameters of the point where the ray intersects, transform the parametrization into one suitable for coloring, and finally use the associated color.
1. Compute the intersection point in object local space using cartesian co-ordinates as usual.
2. Transform the cartesian co-ordinates into spherical co-ordinates.
3. Drop the radial co-ordinate and map the remaining by modulo calculations.
4. Pick a color due to the 2 modulo values.