There is no way that doesn't use lots of graphics primitives.

Instead of using disks, you can build it from polygons in a simple way using something like:

You can visualize the polygons by adding the option Mesh -> All and you can control how many there will be using PlotPoints and MaxRecursion.

One reason why it's necessary to have lots of polygons, and e.g. constructing the disk from "slices of pie" won't work well, is that a triangle Polygon[] can only interpolate linearly between the colours assigned to its vertices.  If you need a colour function which goes through several colours, e.g. Blend[{Red,Blue, Green, Orange}, #]&, then it's necessary to have enough subdivision so all the intermediate colours will have a chance to appear.

If you change Mint to Rainbow above, you'll be able to see what I'm talking about: the structure of the underlying grid will be visible because there isn't enough granularity to smoothly go through all the colours in this colour function.  You'll need to increase the PlotPoints to get a smooth look.

Hi Cesar,

this is not elegant, but the problem is that Disk does apparently not allow for textures. But this works: 

This is the result:



The code is longer than yours, but this allows easily to use any image as texture and the idea is quite simple. If you use yellowtones instead it looks like that:



With other textures one can get:


M.