I always make my pizzas on the grill. I use a pizza stone on top of the grill-grate, but you can also put the dough directly on the grate if you want to. The heat will firm up the dough fast enough that it won't fall through.
The NY style dough recipe on this site looks very similar to the recipe I use, so I think it should work out fine. If you use a stone just top the pizza and proceed as usual. If you put it right on the grate you'll need to follow a different procedure. Stretch the dough and place it directly on the grate, wait approx 3 minutes for it to firm up enough to flip it over with tongs. Once you've flipped then you can sauce and top it right on the grill. Close the lid and wait till the cheese melts and gets a bit bubbly or even lightly browned.
Either way, make sure your grill is pre-heated and is going at full power so that it is hot enough to give you a delicious crispy crust.
Hope this helps.