I don't know about distilled water, but I always use tap water for washed curd cheese without a problem. I have municipal water, which sometimes includes treated well water, and it is safe to drink... I do run it through a filter, just to remove some of the potential for chlorine residue (or whatever else it is they use to sterilize the water), because sometimes I can smell the chlorine scent.
The whey always goes up in pH after washing, so I do not bother to measure right away.
Also, I think I read somewhere that it is not a good idea to use distilled water with a pH meter. No idea why...