No, that's a bad idea. Why would they put their own product against itself? Even if it's for Bragging Rights it's a bad idea because the brand rivalry should remain in the form of one show thinking it's better than the other and vice versa.... having Raw and Smackdown go up against each other would be taking this a little too far. Plus, it would mess up the schedules.... the Smackdown shows would all have to shift to mondays AND become live, neither of which will happen.
It would still be a bad idea. Why put your own product up against itself when you can just give the fans two shows a week like they already do? Raw on mondays + Smackdown on fridays will always get far more impressive ratings than the two of them against each other would since the fans are smart enough to know that it'd be the same federation competing against itself.
Assuming this DID happen.... I would watch Smackdown because it has better matches and no guest hosts. Raw has more entertaining segments, but Smackdown will always get my vote over Raw as long as the annoying guest hosts are around.
There would be no impact on the quality of the shows because Raw and Smackdown are part of the same federation, any ratings earned by either show all go to WWE anyway, so they would put on the same quality of show they normally do. The only way a ratings battle will affect the quality of the show is if the opponent is not part of the same organization.