Why didn't the democratic ideas of greek city-states spread throughout the empire of alexander the great?

Because Alexander the Great was a king, he was a conqueror and was not open to a democratic way to power.

Alexander III the Great, the King of Macedonia and conqueror of the Persian Empire is considered one of the greatest military geniuses of all times. He was inspiration for later conquerors such as Hannibal the Carthaginian, the Romans Pompey and Caesar, and Napoleon. Alexander was born in 356 BC in Pella, the ancient capital of Macedonia. He was son of Philip II, King of Macedonia, and Olympias, the princess of neighboring Epirus.

Once he ascended on the Macedonian throne, Alexander quickly disposed of all of his domestic enemies by ordering their execution. But soon he had to act outside Macedonia. Philip’s death caused series of rebellions among the conquered nations and the Illyrians, Thracians, and Greeks saw a chance for independence. Alexander acted swiftly. He forced his way into Greece despite the roads leading to the country being blocked by the Thessalians. As soon as he restored Macedonian rule in northern Greece, he marched into southern Greece. His speed surprised the Greeks and by the end of the summer 336 BC they had no other choice but to acknowledge his authority.

Believing that Greece would remain calm, Alexander returned to Macedonia, marched east into Thrace, and campaigned as far as the Danube river. He defeated the Thracians and Tribalians in series of battles and drove the rebels beyond the river. Then he marched back across Macedonia and on his return crushed in a single week the threatening Illyrians, before they could receive additional reinforcements.

But now in Greece, upon rumors of his death, a major revolt broke out that engulfed the whole nation. Enraged, Alexander marched south covering 240 miles in two weeks and appeared before the walls of Thebes with a large Macedonian army. He let the Greeks know that it was not too late for them to change their minds, but the Thebans confident in their position called for all the Greeks who wished to set Greece free to join them against the Macedonians. They were not aware that the Athenians and the Peloponnesians, stunned by the speed of the Macedonian king, quickly reconsidered their options and were now awaiting the outcome of the battle before they make their next move.

Alexander's general Perdiccas attacked the gates, broke into the city, and Alexander moved with the rest of the army behind him to prevent the Thebans from cutting him off. The Macedonians stormed the city, killing everyone in sight, women and children included. 6,000 Thebans citizens died and 30,000 more were sold as slaves. The city where Alexander's father was kept as hostage for three years, was plundered, sacked, burned, and razed to the ground, just like Philip acted with Methone, Olynthus, and the rest of the Greek cities in Chalcidice. Only the temples and the house of the poet Pindar were spared from destruction. This was to be an example to the rest of Greece and Athens and the other Greek city-states quickly rethought their quest for freedom. Greece remained under Macedonian rule.