No, each offensive team brings their own. They just have to meet regulatory specs, size, weight, etc.
Granted, this seems like a petty offense, but if you are on parole, the smallest offense has pretty serious consequences. I think this should be viewed in the same manner. The Patriots are rotten, cheating bastards, and repeat offenders.
There is an easy and cheap way to fix this though. Officials should bring the balls from now on and they should be picked randomly from the same bucket... everyone forced to use the same exact ball. QB's will just have to train and get comfortable with the style chosen.
Just do it like baseball. Have a couple of hundred balls all fresh from the factory and inflated to specs and then replace the football after every down. Then sell the footballs after the game for double what they cost.