I tried it on my Genesis but it didn't turn out all that well. The crust cooked way too fast and burned even before the cheese melted. I think it would've turned out better if I elevated the pizza stone up away from the grill grates instead of placing directly on the grates.
But then again I never really had success using my home oven either. I'm not a pizza master or anything but I did own NY style pizza restaurant before. I could get perfect pies using the store brick oven but at home it's worse than any delivery pizza chain. So I don't make pizza at home. I make calzones instead and those always turn out perfect.
Mine was bad, the stone was so hot the cornmeal I put on it so it would slide easier burnt up before I could get the pizza on it, then the store bought dough would not cooperate, remained too thick and the end result was a badly burned bottom and a doughy chewy top, it sucked. Maybe my stone needs washing but everywhere said do not use detergent on it, fuck it, I went to Wendy's. I can cook a lot of tasty meals but I'm done with homemade pizza forever..