Hi, I went here: https://us.battle.net/account/download/ and there is no option to download Diablo 3? only games I see there are: World of Warcraft Starcraft II Classic Games where do i download Diablo III?
if you have bought the game registerd it to your battle.net account it should add the game to your list and also give you the option to download a digital copy but you must activate your product key