Sparta was not part of the dominion that Alexander inherited from his father, for reasons I outlined in detail in an older post. This meant it was wiser for Alexander to leave it alone.

In the early years of his reign, he had his hands full asserting his rule over rebellious parts of the Macedonian sphere of influence, always careful to act as a magnanimous overlord to the loyal and a ruthless avenger to the disloyal. An attempt to conquer Sparta would distract him from this goal, and likely only worsen his crisis of legitimacy. It would amount to an unprovoked war of aggression against a neutral state that had little power to undermine his rule.

Alexander also immediately made it known that he intended to reunite the Greeks behind his father's plan for war against Persia; waging war against Sparta would, again, only work against him, by making him seem like a violent conqueror of Greeks rather than a righteous crusader for the Greek cause.

In short, even though he could have conquered Sparta, it would have been diplomatically disastrous to do so. Sparta was not dangerous; it was encircled by its own bitter enemies, all of which were subject to Macedon, and little could be gained from trying to complete the conquest of Old Greece.

Instead, Alexander continued his father's policy of letting Sparta ruin its own reputation by refusing to join his campaign. After his first successes in Asia Minor, he famously erected monuments and sent spoils to Greece that were expressly dedicated by "Alexander and all the Greeks except the Lakedaimonians" - a clear message that there was only one Greek state arrogant enough to refuse to join the obviously righteous and long overdue war to break the power of Persia.

Sparta did eventually become a problem for Alexander, when it organised an alliance of Peloponnesian states to rebel against Macedonian rule in 331 BC. This rebellion was allegedly funded by the king of Persia (or at least unsuccessfully applied for funding from the Persians), since it was hoped that it would prompt Alexander to withdraw to Europe and deal with the rebels. But Alexander would not be distracted; he sent his regent Antipater the sum of 3,000 talents (enough to pay 50,000 mercenaries for a year, if necessary) and told him to handle the situation. Antipater proceeded to gather an army of Macedonians and Greeks and crush the Spartan-led insurgents at the battle of Megalopolis. Alexander reputedly joked that, compared to what he was doing himself, the war with Sparta was like fighting against mice.

In the aftermath of the rebellion, the Spartans sent envoys to Alexander to beg for his forgiveness, which was apparently granted. Some of the actual rebel subjects were fined, but Sparta was not subjected to Macedon.