I'm sure RBH didn't mean specifically just the "Screwjob." He meant the poor situation and relationship between Bret and Vince that led to Bret's departure.
Well, anyways to answer the question, it's hard to say. To start off with, Bret wasn't very thrilled with the direction that the WWE went in around 97. I don't think he would've been so against it that he would've left, but who knows? By 98/99 when it was far more edgier and racier than it was in 97, he may have left?
Then again if Bret hadn't left, he was such an important and influential figure in the WWE, that he may have been able to influence some of the direction that Vince went with the WWF (much as HBK and Triple H did), and maybe it wouldn't have went as hardcore as it did.
But I think Bret would've stayed for sure, at least until May of 99 and the tragic death of Owen. Now if Bret was with the company, as he has stated many times, he would've seen to it that Owen would've never done the stunt. But if he somehow didn't stop it and Owen did the stunt and died, that would've definitely been the end of Bret's relationship with the WWE and Vince. Ultimately it would've tore him up much more than the Screwjob did.
But....if the accident never did happen, Bret probably would've went on wrestling, perhaps into today. I think if all of the tragic events that have occurred for Bret in the last decade (the screwjob, Owen's death, his career-ending concussions, and ultimately his stroke, not to mention the drama he's had with his family) hadn't happened, then Bret would probably not be trying to move on from the last 10 years like he is currently. He would still happily be in the wrestling business.
Bret always said that he never wanted to leave the WWE. And I think ultimately even though he was not a supporter of the more "Attitude Era"-styled product, he would've stayed. Especially if everything with Shawn Michaels went as it has. If Shawn would've suffered his back injury and retired after Wrestlemania XIV, the whole "The WWF is too big for Shawn and Bret both" problem would've been solved, and Bret would've been more content in the WWF without Shawn there.
As long as the accident with Owen would've been stopped, Bret might still be wrestling today, as long as he was healthy and didn't get injured. Bret would've enjoyed wrestling the new guys that came into the company from 98 on (Edge, the Hardyz, Angle, Tazz, Jericho, Big Show, Christian, Benoit, Guerrero, Malenko, Mysterio, Lesnar, Cena, Batista, Orton, etc.)
And I don't think you have to worry with Bret still being there, would Austin and the Rock have made it to the top? They would've. Bret would've stepped aside to let them two and Foley and even Triple H (though he would've hated it
shine. Actually the plan originally was for Bret to fight Austin at Wrestlemania XIV and lose the title to him. Bret would've had no problem putting the new guys in the company over.
It's a real shame things didn't turn out differently.