The molecules that make up most white things usually aren't able to screen (what a metal does) or significantly absorb light with energies high enough to be damaging to DNA. What you see as white is usually only good at scattering most might in a relatively narrow band of visible light, and since scattering doesn't change the angle of the main energy path much, it's not really protective (you'll get a chain of scattering off of lots of molecules as the photon dumps energy into them, including damaging the DNA).

Similarly, absorbing light is difficult as the higher frequencies require electronic and eventually nuclear transitions (this is the difference between x-rays and gamma rays) to actually absorb the photon. So scattering is the predominant energy absorption mechanism for biological interaction with gamma and x-rays.