the simpliest answer is that it made more economic sense to expand his power towards conquering the rich and advanced civilisation of Persia, and then eastward towards Tigris, Euphrate, and Indus: the craddle of civilisations, and places with significantly greater riches, greater populations, and more advanced economies than comparatively tiny Greek poleis and fledgeling Rome.

Sources: (among others)

Heckel, Waldemar; Alexander the Great: A New History.

Green, P, Alexander of Macedon, 356–323 B.C.: a historical biography,

As u/freevoulous has taken care of the economics and empire building, I give you the politics and the personal:

Alexander's father, Philip II of Macedon, had schemed and conquered his way to the top of the Greek world with a view to rolling back Persian conquest in Anatolia and had played up the avenging of historical humiliations (much of the Greek world had been forced to accept vassal status to stave off invasion) and the reconquest of the historically Greek communities along the coastline of Asia Minor.

When Philip II was assassinated, Alexander inherited a force that was already in war footing, poised to cross into Asia. Momentum was already in place and there was something of a political imperative that he follow through.

As to rolling on Eastward, self-aggrandizement may have initially played a part. Macedon was perceived as a barbarous hinterland full of "uncouth" tribesmen - well outside of 'classical Greece' - and Philip had humbled Athens and Thebes. This was still clearly the case even in the reign of Alexander, as Herodotus seems to put a telling amount emphasis on how Hellenic they are ("And that these descendants of Perdiccas are Hellenes, as they themselves say, I happen to know myself, and not only so, but I will prove in the succeeding history that they are Hellenes." - jeez, we get it bro, they're 'real Greeks').

As much as we can say anything conclusive about him based on writers from later centuries, his ego was considerable - even early in the campaign, before he started alienating his loyalists and depicting himself as Emperor, he made a great play of presenting himself as the heir to Achilles. His first 'Alexandria' - Alexandropolis Maedica - was founded when he was only 16 after suppressing a revolt on his father's behalf.

Further to the case for ego, rather than turn east toward the Persian heartland after Anatolia fell, he turned south-west. The Ancient Greeks viewed Egypt as the cradle of Mediterranean civilization and learning and he literally built himself into that landscape in the form of the City of Alexandria and Special Edition-ed himself into the pantheon as the son of Amon-Ra. Now a Pharaoh, it was onto Babylon and Persia, to set himself up in the finery of the deposed King-of-Kings and more ruined cities of past glories.

Sources: Alexander the Great by Robin Lane Fox and In the Footsteps of Alexander the Great by Michael Wood

My sources (in Greek language) mention that Alexander had a mere 800 talanta readily available. He needed another 1200 talanta in order to make a considerable army and navy and maintain it for 3 months (maybe 4). He had to act quickly and conquer Sardis, where he indeed found 10000 talanta, enough to pay loans off and feed the army for a couple of years (not mentioning the taxes and tolls he would get). He got the initial 800 talanta from his family property, and other noble Makedonians contributed. The rest he got it from banks (temples or other rich people inside and outside his core kingdom) in the form of loans.

The following are speculations and open questions. As mentioned, Alexander had the momentum. It was the ideal time, a short time window when he had strategical advandages, including the best open field infantry and open field cavalry of the time. He wouldn't have this advandage for long, innovations tend to diffuse (and horse breeds tend to diffuse too, not to mention generals and mercenaries being lured by money together with their skill). So he had to act quickly.

He destroyed 2 cities after his father's death afaik: Tyre and Persepolis; both the crandles of 2 major civilisations, the king of the east and the king of the west. Some might say that this implied that he would go against Carthage as well. But he had no navy.

Besides navy, he must have taken into account other parameters:

• His army was best suited for fights in plains. If he somehow managed to go beyond Asia Minor, he would find favorable terrain in Mesopotamia or Egypt. He was fortunate that Persians did 2 major mistakes: they did not do guerilla and they kept too much gold in their vaults. If he went to the west he would never find favorable terrain until he reaches France... while at the time all the money were Sicily, Central-Northern Italy, Spain and perhaps some gold-producing zones such as Dacia or Dalmatia. Mountains, piracy, dense and poor populations, not the best plan. Going against Persians he had the chance to end the war quickly if Persians were lured into a set fight. He would need many years if he were to go towards the West, he would waste his temporary advandage.

• Perhaps the failure of Athenians in 417 against Sicily taught him something.

• Not sure how the free cities in southern Italy would see his campaign.

• He already controlled part of southern Greece's grain supply by controlling Byzantium. He needed to cut another 2 sources if he were to use the monopoly: Sicily and Egypt. I bet he wanted to go to Sicily before Rome, but not before Persia.

• Northern Europe indeed has wide plains, he could go to Vienna area or Danube and then further to the north, but (i) his father already has proceeded towards the north and his efforts did not reward him quickly (ii) he could not let Persians intact (and Carthaginians too) for a long campaign towards the north. Furthermore, how would other Greeks react to his abandoning of the anti-persian propagandised goal? Would they sit quiet in his absence? The alternative, the prospect of gold influx from the east, would calm Greeks down for a long time.

Speculations, I would love to hear objections.

Simple Version: The plans to invade Persia where made before Alexander came to power.

When Alexander came to power, his father (Philip II of Macedonia) had, before his untimely assassination, already been appointed leader of a campaign among the Persian empire. The stated purpose of the invasion was to punish them for their damaging invasions in Greece in years past. After establishing himself as the successor of Philip and new leader of the League of Corinth, the league which Philip had formed including the city states of Greece under Macedon, he also took charge of the campaign. It simply made more sense to invade Persia. The campaign was already in its early stages, the army was ready, the plans were basically drawn and Greece was more or less behind the idea already.

However, after securing the Persian empire and beyond, Alexander seems to have planned to then go west. In his testament written some time before his early death and read aloud after it, he called for the conquest of the southern and western Mediterranean and the control of the entire Mediterranean basin, which would include some of the areas you mentioned. So basically he would have gone their, once he was finished in the east - or, as finished as Alexander the Great could be when it came to conquering.

Perhaps the OP, by his phrase "expanded his empire eastward", meant Why did Alexander continue expanding eastward AFTER conquering the Persian Empire?

Many previous posters have shown that Alexander and his army was primed for a war with Persia, both militarily and by propaganda.

But once he defeated Persia, the Persian king Darius fled all the way to Parthia. This alone takes him well into Western Asia, past the Caspian Sea. Darius was then taken hostage by Bessus, a Bactrian satrap. As Alexander approached, Bessus had the king killed and promoted himself as the successor to Darius and therefore to the Persian Empire. Bessus fled into Central Asia, where he would engage in a prolonged and costly guerrilla campaign against the Macedonians.

Alexander couldn't abide this usurper to the throne, and deemed necessary an expedition to suppress Bessus and his loyal forces. This expedition turned into basically a general Central Asian Theater opened in his war, bringing him through all of what we today know as Afghanistan and Tajikistan.

When, after this, the Sogdians were raised in revolt and yet another punitive campaign was launched against them, Alexander and his forces were right at the foothills of the Himalayas, his empire butting up against Northwestern India. As another poster has said, he could have turned back westward, but in doing so may have risked the dissolution of much of his army -- the main campaign against Persia now finally completed, after much digression. And then, also as previously stated, there was Alexander's ego: India was basically the end of the known world, at the time -- how great and tempting a prize must it have been to just "push a little bit more east" and gobble up that immensely rich and populous country, and include that exotic gem in his trove of conquered empires and peoples...