This is an old theory known as the Grand Hoplite Narrative (or "hoplite revolution theory"). Roughly, it argues the following:

• In the late 8th century BC, the Greeks adopted a new double-grip shield
• This shield was most effectively used in a densely packed formation called the phalanx
• The rise of this formation made the 'middle class' owners of hoplite (heavy infantry) equipment militarily more significant than the old aristocracy (who were too few to make up a phalanx alone)
• Aware of their new collective importance, the hoplites started demanding political influence
• Bam! Democracy.

For this theory see in particular the works of V.D. Hanson, Donald Kagan, Kurt Raaflaub, Paul Cartledge, Gregory Viggiano, etc.

Recent scholarship has torn this theory to shreds. For one thing, there were many other kinds of state organisation in Archaic Greece; many city-states were not ruled by the people, but by a select oligarchy or a tyrant. They frequently switched back and forth between different forms of government. In fact, there is no evidence of any state going neatly through the stages of adopting hoplite equipment, having a popular revolution, and becoming a democracy. This process is 100% theoretical, based on some speculation by Aristotle that this is what may have happened at some ill-defined point in the past (Arist. Pol. 1297b.15-34).

For another, it cannot be shown that the phalanx was introduced that early. The earliest literary description places it in 480 BC and the word in its technical sense doesn't occur until the 4th century BC. Authors like Peter Krentz, Hans van Wees and Louis Rawlings have shown that Archaic Greek combat was much more fluid, with different troop types mixed on the battlefield. The introduction of the new shield probably did not lead to any major change in the way battles were fought, which means it cannot have led to any resulting change in social hierarchy.

More fundamentally, there is no evidence that the hoplites made up a distinct new social class. In the Archaic period, the hoplites didn't fight the rich; they were the rich. They were a small upper level of society and they continued to be in charge just like they had been before. In later times, the hoplites included both the rich and the poor - anyone who could just afford to buy the equipment could qualify. The idea that the hoplites formed some sort of previously unrepresented social stratum is, again, completely theoretical; there is no evidence to support it.

Nevertheless, because the theory so nicely ties everything together, and because it gives great importance to a supposed 'middle class', a lot of scholars (especially in the US) refuse to let it go in light of reasonable criticism. This is why it still dominates a lot of entry-level/textbook/popular history treatments of Greek warfare.

Edit: TL;DR: No.