In a medium bowl, stir together the tomatoes, 1/4 cup of water, olive oil and the salt. Adjust the seasoning if necessary.
Preheat a grill to medium-high. On a lightly oiled, inverted baking sheet, pat out 1 piece of pizza dough with your fingers into a free-form shape about 1/4 inch thick. Lift and transfer the dough onto the grill and cook with the grill cover down until the crust holds its shape and grill marks appear, about 3 minutes.
Using a spatula, flip the crust over onto the coolest part of the grill. Spoon a couple of tablespoons of the tomato sauce over the dough and scatter with about a 1/4 of the mozzarella. Cook with the grill cover down until the cheese is melted, about 3 to 5 minutes. Remove the pizza from the grill and top with basil leaves. Repeat with the remaining dough and toppings.