Preheat oven to 350 degrees F (175 degrees C). Spray the bottom and sides of a 9x13-inch baking dish.
Bring water and rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender and liquid has been absorbed, 20 to 25 minutes. Transfer rice to a large bowl. Add diced tomatoes with green chile peppers, garlic salt, onion powder, chili powder, salt, and pepper and mix well. Spread rice mixture into the bottom of the prepared baking dish.
Layer chicken tamales on top of rice mixture. Mix sour cream and salsa verde together in a bowl; pour over tamales. Sprinkle Cheddar cheese and Monterey Jack cheese over tamales. Cover dish with aluminum foil.
Bake in the preheated oven for 20 minutes. Remove foil and bake until cheese is browned, about 10 minutes more. Let sit for 10 to 15 minutes before serving.