Well likely Darius’ reputation was pretty much ruined after the previous battle of Issus, when he lost the battle and escaped leaving his family inside his camp but still at least he escaped only once instead of twice and was killed in the battle, not by his supporters during the flight...
Well if he has no need to go at the eastern border is pretty easy finding something else to conquer (maybe Arabia and after consolidating his power in Asia, likely with earlier weddings of Susa, he will go to conquer Carthage, Sicily and avenge his uncle and brother-in-law in Italy)
see below
Darius III definitely lost more prestige by fleeing constantly and especially leaving his royal camp to be captured. In the legitimacy system he lived under, that Elamite-Assyrian-Babylonian understanding that Persian royal legitimacy was based, such an action is unforgivable. Many Assyrian kings were defeated and some killed in battle. These were not bemoaned as failures. However, an Assyrian king who avoided a battle or fled without a tactical cause, was dethroned. Persian royal understanding was similar to this, at least in its early days.
That's why I said Darius would likely recover some of his reputations here, and there are arguments that his defeat at Issus is much more tactical and through compared to Gaugamela (where Darius actually fled when his right flank could be said to be actually winning).
Darius' defeat at Gaugamela was caused mostly by the Psychological impact of him fleeing too early. Have him stand his ground, even if he died in the process, there are possibilities that the left flank under Parmenion will actually break, as the left-wing Macedonian cavalry has been encircled by Persian right-wing. And if Darius didn't died on impact but fight on for a while, there could be the chance that the Elephants would be used to break Alexander's companions instead of merely sitting their asses at the rear.
But yeah, since the WI was about Darius being killed, not *successfully* standing his ground, I'll assume he put up a fight, managed to use his Elephant forces to maul the Hetairoi, and overall, Alexander's victory at Gaugamela could be called as Pyrrhic (if not because Pyrrhos isn't even born yet). Instead of emerging victorious with his forces mostly intact, Macedonian Left Flank cavalry would be mauled by Persian cavalry before the later eventually broke, and Parmenion is likely to be dead from such fight. The Pezhetairoi at the center might be spared from too many extra casualties, but I would bet that the elite Hypaspistai and Agrianikoi would be mostly slain here.
In essence, Alexander would still win, but this time, he would be forced to actually consolidate the territory he has conquered, and wait for another reinforcement to come from Makedonia before he could hope to press further east. He might be able to take Babylon, but he would need to essentially forced to recuperate his forces there.
In the meantime, maybe Bessos or Ariobarzanes could use the precious time Alexander spent recuperating to muster a much larger force at the Persian Gate, this coupled with the mauling he received at Gaugamela, might force Alexander to actually reconsider his options of consolidating the gain instead of pressing further east (it's unlikely Persian remnants could actually attack Alexander's position either).
Alexander might still want to go east, but it would take at least another two or three years, enough time to get busy with his wives and getting Alexander IV early on. Plus, with Parmenion died "Heroically" at Battle, there will be no need to purge him later, that means if the poisoning plot was true, the plotters won't have any reasons to kill Alexander this time, so Alex might live much *MUCH* longer here.
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And as for Darius, sure, he would still be shamed on fleeing at Issus, but once the records of the battle become known, it would be certain that Darius was FORCED to flee or face certain, needless death (because his isolated force of chariots was bogged down and Alexander led the Hypaspistai attacking ON FOOT). He would be then lionized at Gaugamela as he heroically stands his ground there. Some accounts might even romanticize the depiction by writing about Darius and Alexander facing each other in single combat or something like that, considering Darius' own youth, it was a believable story.
