I bought Diablo III from Blizzard. Why won't it let me re-download it on another computer?
You should be able to download the game and install it on as many computers as you wish. In fact, since your game purchase is tied to your Battle.net account, you can download the game even if you haven't bought it yet.
This post has a list of links to various versions of the game's installer. You can also re-download the game by logging in to your Battle.net account and clicking the "Download Game Clients" link.