There are two PODs that have essentially the same effect. One is that Alexander dies just after his old man, as he is consolidating his position in Greece and Thrace (which he had to do before embarking on his expedition). Macedon collapses into in-fighting, and the practical plan to invade Persia was very much Philip's idea. The general notion still exists, but without Macedon to spearhead the effort, the Greek city-states (happy to be freed of Macedonian overlordship) are very likely delayed in actually carrying it out.
The other POD is that Kleitos the Black is a few seconds later, and Alexander gets a second axe-blow to his head at the Granikos-- shattering his helmet and killing him on the spot. His campaign collapses into disarray, and is easily mopped up by the Persian forces sent to deal with the problem. In this case, as well, Macedon collapses into in-fighting. The Greek city-states consider themselves feeed from the Macedonian yoke, and any renewed Hellenic expedition to the east will not come about for some time.
And that has consequences, because...
Alexander wasn't a matter of luck, he was a perfect storm of circumstances.
...circumstances will change over time. And soon! The Achaemenid empire was having trouble at the exact time Alexander invaded, but was by no means moribund or doomed. Given a few years to recover from the effects of a its recent succession struggle, it would be in a far better position to fend off any Hellenic invasion attempt. That said, certain areas in the far west were hard to defend against the Greeks; Egypt and western Anatolia might well be wrenched loose from Persian control. But I don't see more than that just happening. (Of course, it is
possible that the Achaemenid empire just declines for unrelated reasons anyway, and becomes easy prey anyway. But that's somehing the writer of a TL has to make up. For the purposes of this scenario, let's assume that doesn't happen.)
So what's the general situation in Greece? Well...
The Greek polities in the era of Phillip of Macedon were forming larger and larger leagues. Their population was booming, their organizations becoming more federalized or centralized and more advanced.
...this is very accurate. But this...
Sooner or later, the peninsula would be united, and sooner or later that unified polity would need to strike against Iran, both for its own expansion and to secure it's long term stability in the face of Iranian pressure and attempts to disunify this new rival. Accordingly, you'd see some sort of major invasion by a Greek polity into Iran.
...might not be. It's just as likely that Greece, freed from the Macedonian overlordship established by Philip, and no longer facing a Macedonian threat (becaue Macedon will have collapsed into mutually hostile fiefs, as it always threatened to do before Philip consolidated his power), doesn't unite. The Greeks didn't particularly
want to unite. A system of competing leagues is more likely, in my opinion.
As I pointed out, the Greeks might well try (and very possibly succeed in) grabbing Egypt and (western) Anatolia. Possibly, one league grabs Egypt, and another goes for western Anatolia? This scenario, of two or more competing leagues, seems like a very plausible outcome to me. More likely than Greece being unified as one whole. The Persians, of course, would try to play off the Greek polities against each other-- as had always been the Persian strategy.
This means that the following...
The Era of city states is over, but the manpower of the Greek world isn't drained. They'd have to turn somewhere. We'd see adventurers in Carthage and Italy, Egypt and Sicily. We'd see the young men of Greece going abroad for opportunity, and attempts by these federated Greek states to exert their power across the Mediterranean.
...may not be the natural outcome at all. Greek adventururs will end up in Egypt and Anatolia (which are in any case much richer than the west), and iof exerting power, they're most likely to be duking out their rivalry with
other Greeks. The Persian aim will almost certainly be to cause another Peloponnesian war, pitting Greek against Greek so the threat to Persia is much reduced. that strategy has worked before, and might well work again. Even if not, I don't really see a single Greek 'empire' taking over the westwern Med. Personally, I thinbk it's very likely that such a strategy would work, and that the Greek world would end up in a war against itself anyway.
The rise of Rome would not automatically be butterflied. All this is very much based on choices made by a TL's author. Everything can work out in various ways... but it's not a given that Rome would be butterflied, or that it would be facing a strong and united Hellenic world. Rome might still end up conquering the Hellenic world, much as it did in OTL.
At that point, the question becomes... what is Persia like at that point? Well...
And if the Romans reach Persia, what reason is there that the Achaemenids are still in power? A Saka dynasty might have conquered them, or any number of resurgent threats. Undoubtedly their army would not closely resemble that which fought Alexander. It would be newer, adapted to fighting disunited Greek states and the Saka alike. Expect a far more deadly cavalry arm and a much greater reliance on heavy infantry forces. Expect an Iran that's well accustomed to being an imperial power, and could probably provide a plausible threat to the Romans.
...something like this is very possible. But for the sake of the scenario posited in this thread, let's assume it's not the case.
Realistically, I just don't see Rome conquering all of Persia. Not even if Persia has, over time, really become a moribund empire ripe for the picking. As pointed out, Alexander was the 'perfect storm'. That kind of conquest, in one fell swoop, only happens whwen it is driven by one personality bent on achieving it. The Roman Republic didn't work that way, really.
On the other hand: military success was one way to gain political status in Rome. One could imagine a very capable general conquering Persia almost by accident. Going for the near east... and just... succeeding beyond his own expectations, and pusing on. And on. And on.
Even in that case... I don't see Rome
holding it. It's too big, and their centre of power is too far away. Most likely, they'll try to form allied states in the region, while annexing everything up to Mesopotamia and Armenia directly.