Assuming that neither army would have been driven from the battlefield a new set of options would have been presented. One, Rosecrans withdraws his damaged army back to Chattanooga within a day or two. Two, Bragg withdraws his damaged army to Dalton within a day or two. Three, the battle renews the next day for a third day of fighting. All of these options are plausible. The second option, however, seems to be the more likely event. Bragg has a history of retreating following an indecisive battle. He retreats following Perryville and Stones River and neither battle would have been described as a resounding Union victory prior to Bragg's retreat. It is only with Bragg’s withdrawal that they became Union victories.
If Bragg did retreat again it can be assumed that Rosecrans would wait until the spring to begin a campaign in northern Georgia. Rosecrans was a cautious commander and the large number of casualties at Chickamauga would require time to replace. Rosecrans would also likely build Chattanooga into a supply base much as he did with Murfreesboro in the months after Stones River. Another possible outcome of this battle might be that Bragg would be removed from command of the army and that Longstreet might take his place. That is just a guess though and its impact on the future of the war is anyone's guess.
If it is decided that the Dyer Ridge defenses would have played a key part in saving the right it can also be argued that perhaps the Union left would now have too few troops to hold that area. Changing one aspect of the battle would have effects on other areas and it is possible that this change might have prevented the rout of the right wing but still lead to Union defeat, however I do not believe that Thomas would have been driven from the Kelly Field position. The troops that would have stayed on the right flank were troops that Thomas either never saw or that he used to defend Snodgrass Ridge. Thomas would not have to defend Snodgrass Ridge if there had been no rout of the right wing. The last reinforcement Thomas received prior to the breakthrough was Van Derveer's brigade and this he certainly did need. After that though Thomas is not faced with another crucial moment except on Snodgrass Hill. If Thomas does not have to defend Snodgrass Hill then he should have been able to hold Kelly Field. Possibly Bragg would have tried to find troops to make an attack on Thomas' left flank if Longstreet's assault was a failure but that is another guess that is impossible to prove or disprove.
If Chickamauga had been a Union victory its impact on the war could be as varied as the possibilities afforded by the Dyer Ridge defenses were to the battle itself. In the east two corps would never have had to leave the Army of the Potomac to aid the Army of the Cumberland at Chattanooga. A winter campaign against the depleted Army of Northern Virginia might have achieved some successes.
In the west Major General Ulysses S. Grant would still probably be promoted to command the Military District of the Mississippi but a promotion to General-in-Chief and Lieutenant General might not happen in the spring. Rosecrans would probably be re-enforced with some men to make up for the losses of Chickamauga and also to offset the Army of Tennessee’s gain of Longstreet’s corps. Where would Grant command the district from? Would Grant command in the field with Rosecrans, prodding him along as he did with Meade? Where would Grant put Major General William T. Sherman’s army? Sherman operating against Mobile while Rosecrans goes after Atlanta might have hastened the end of the war in the west. If the Army of Tennessee keeps Longstreet’s corps the capture of Atlanta will likely be harder to accomplish. If Atlanta is not captured by the fall elections will Abraham Lincoln be re-elected? In Virginia though the Army of the Potomac should find itself with an easier task and so perhaps Richmond is captured before Atlanta is.