Why is the moon light white?

The tendency for the peak luminicense sensitivity of the eye to shift toward the blue end of the spectrum at low illumination levels as part of dark adaption. In consequence, reds will appear darker relative to other colors as light levels decrease, Simply the reason is Purkinje effect.

Source : pixabay.com

The moonlight consists of sunlight and earth light reflected from it's surface which is of low intensity levels. So, what we usual see is bluish(on full moon days) or most of the times moonlight is referred to as silvery. If we observe the color of moon for lunar eclipse, it appears as red/ orange because it's indirect sunlight refracted and scattered by the Earth's surface.