Quite simply- logistics. Arabia is doable, but much beyond that, I think Alexander's going to struggle, and get bogged down with revolt and rebellion in the Iranian highlands and elsewhere.
I'm not really sure why this would necessarily be the case. Several of the Iranian provinces revolted against him in India because the satraps he reappointed thought that he would never return from India and wanted to be ahead of the game. When Alex returned, he got rid of the troublesome old satraps and appointed his own new guys, guys that were far more loyal to him. Peucestas is probably the best example - he was very loyal to Alexander and had earned the respect of his constituents by learning their language and adopting their customs. The Persians liked him (and I should note that Persia was by far the trickiest province for Alexander or his men to win support), and he was loyal to Alexander - a recipe for stability, not rebellion.
When Alexander died, the empire initially stayed largely intact. I can only really think of two rebellions (that weren't led by one of Alexander's ambitious generals) immediately after Alexander's death: Athens' Lamian War, funded by Harpalus' talents, and the Greek settlers in Bactria, who wanted out and only stayed there because Alexander forced them to. Both proved manageable. Even the Indian provinces remained with Perdiccas and the new regime, until Chandragupta conquered them. Iran remained wholly under Macedonian control for another couple hundred years, even during the chaos of the Diadochi wars, and even after Seleucus Nicator's death. Only the Parthians, a province that the Seleucids neglected while warring constantly with the Ptolemies over Syria, could get the Iranians to revolt against the Macedonians.
During the three years or so while Alexander would campaign in the west, he would probably have a trusted regent (someone like Lysimachus) back in Babylon to deal with potential rebellious satraps in Iran, India, or Arabia. If one were to revolt (which obviously I think is a lot less likely then most here do), Alexander's crony in the east would very likely be able to take care of it. They wouldn't be likely to revolt en masse, because these satraps are 1. far more loyal to Alexander then their predecessors and 2. not willing to chance that Alexander would die in the west like the previous Iranian satraps were to chance that he'd die in India.
To sum: satraps are Alexander's biggest concern in maintaining his empire during campaign, and after his war in India, he has managed to appoint satraps wholly loyal to him (or at least far more so then Darius' old men). Mass Iranian dissent against Alexander is unlikely, because of men like Peucestas and the lack of a nationalistic feeling for Iranian peasants (who don't particularly care who their rulers are, as long as their new rulers don't make a point of pissing them off by raising taxes or whatever - they might prefer Iranian rulers, but aren't willing to rise up and force the issue because their lives are essentially the same). The Greeks (read: Athens and Sparta, and possibly those Greeks in Bactria) are his biggest rebellion concern, a concern he started to deal with with the Exiles' Decree, which could only help him win support in Greece.
Essentially, an even more summed up version is this: why are satraps that Alexander appointed revolting against him, when they didn't against Perdiccas?
Kind of ironically, it seems to me that the empire was most stable right around the point that Alexander died. It broke up because of infighting between the generals once his presence was gone, not because the east was mad with Alexander or anything (well, maybe Chandragupta, but besides him!). With Alexander still alive, his absolutely enormous presence would still be around. He was one of the very few people in history that could keep together an enormous empire by sheer force of personality... and healthy paranoia/political sense (whichever you want to use
), which eliminated many potential rebels before they could get far.
As to how the western campaign would go, ignoring events that may or may not be going on in the east during such a campaign... there are two ways that he could approach it: taking the "Pyrrhus" route, or taking the land route from Egypt all the way to Carthage itself. I personally think that the Pyrrhus route is more likely - that way, he can conquer the Samnites and Lucanians (the ones bothering Tarentum), and he can gather Greeks along the way, and ensure Magna Graecia's loyalty by garrisoning the necessary towns. A Siege of Lilybaeum would be tough, but doable with Alexander's massive fleet that would accompany him. Then, he'd probably incite rebellion in Carthage by getting the Libyans to revolt and by gaining the loyalty of some Numidian princes. A pitched battle with Carthage would be fought that Alexander would win easily due to numbers as much as anything, and then a tricky, lengthy siege. The western Punic colonies would submit during or immediately after Carthage's fall... and afterwards, he may or may not want to go all the way to the Atlantic with his army. Then, the return home... home meaning Babylon, of course!
