Yep, if you skim you will get a whiter cheese. The color is all in the cream. Commercially, a fuller-fat mozzarella is made by separating the cream and then blanching out the color through heat (carotene is heat-sensitive and breaks down quickly above 200F), and then adding the cream back in, or the processor may add in a food coloring instead like titanium dioxide. I heard CHR Hansen recently came out with a cheese whitener additive, too.
At home, you can also control color through feed. Winter milk produced from dry hay/alfalfa and supplements tends to be whiter.