You're not supposed to wash a pizza stone!
My pizza stone lives in my oven. The heat seasons it. If any stray cheese or whatever gets stuck on it, I scrape it with a metal spatula after it cools a little (cool enough to stick my hand in the oven but not so cool it's solidified). It sure is black and a little splotchy! That's exactly what it's supposed to look like. You can "clean" it better by leaving it in the oven during the self-cleaning cycle. That will get that whole layer of "crust" off. If yours doesn't self-clean or you don't want to use the cycle, you can also put it in a charcoal grill (well, put it in cold and then bring the grill up to a really high temp ... wouldn't want anyone to crack their pizza stone!).
If you want something pretty to cook pizza on, I'd suggest a cast iron pizza pan
. You shouldn't wash cast iron in the conventional sense either though. (My mom had a cast iron pan when I was little, and I always wondered why people loved cast iron because anything you cooked in ours stuck like glue! Turns out you're not supposed to use soap on it. Huh. She didn't know!
If you have used soap on it, you should soak it in hot water and re-season it in the oven after it dries completely, before you cook another pizza on it. The soap can be deep inside and come to the surface (and get into your pizza!) during cooking, since the stone is porous.