I would like to add to "LuckyOne"s solution. As I see it you want the colors to rotate around the sphere origin. The base idea is still the same, make geometry nodes output some values, that you can then use in shading nodes.
Example:


Which produces:

The problem is the points at the top are closer together so they do not follow right colors. (But it is still a nice effect, which might be what you want).
The second way to get a similar result is to create a Cartesian to spherical conversion coordinate system node. This way you can get exact angle from origin which produces the following effect.

The nodes for this example are slightly more complex:

Where the spherical cords is basically just math from https://en.wikipedia.org/wiki/Spherical_coordinate_system. The nodes look like this:

The output of this node is going to be a field of values between -pi and pi, but ColorRamp fac wants values between 0 and 1. So we need to remap this array. With some simple math we can also make an offset, so that we can animate the movement:
