I've already turned Azura's star white and now when I see of it over the Internet it's just been wasted can I turn it black now?
No, once you've finished this quest there's no in-game method of converting the star from one form to another. I wouldn't lose sleep (or significant game progress) as a result of your choice - soul gems are reasonably common, even black ones, so Azura's Star is not particularly essential in either form.
On the PC, you could use the console to give yourself a Black Star:
As always, I suggest backing up your save prior to using the console, as mistakes made with it can sometimes cause damage to your game.