To what extent did the Great Awakening justify the politcal thought of the American Revolution?
The Great Awakening had little or no bearing on the Revolution. There was nothing political about the Great Awakening, other than it did bring about the demise of Puritan ideas of predestination and people being called to their work. Even then, the connection to politics and political thinking is dubious at best. It saw the birth of denominations such as Methodist and Baptist, but other than that had no bearing whatsoever.
The Revolution was based primarily on the denial of the colonists' rights as Englishmen, and further upon Enlightenment ideas of basic rights for all individuals. Enlightenment thinking was anathema to the leaders of the Great Awakening, as it treated God as the great watchmaker who was not involved in people's daily lives. Ministers such as George Whitefield, mentioned above, and Jonathan Edwards emphasized that God was keenly aware of people's daily lives and they could have a personal relationship with Him. This is hardly political or revolutionary thinking; nor does it have the slightest taint of Deism. Thomas Jefferson, who wrote the Declaration was himself a Deist, and objected to the inclusion of any reference to God in the Declaration of Independence. It was added by the Continental Congress over his objection. George Washington, a leader of the revolutionary movement was not a religious person at all; he never kneeled to pray, and only spoke on occasion of "divine providence." None of the great leaders of the Revolution, in fact, had any relationship, direct or indirect with the Great Awakening, aside from the fact that Aaron Burr was Jonathan Edward's grandson.
The above post comments that George Whitefield never went to College; the other great leader of the Awakening, Jonathan Edwards was a Harvard graduate, first in his class. As far as ale houses, John Adams once commented that taverns were places where babies and revolutions were conceived. So, there is a connection between revolutions and ale houses, but not between revolutions and Whitefield's preaching. Whitefield, in fact was British and returned to Britain after preaching in the colonies. His sermons emphasized, somewhat dramatically, the dangers of the fires of hell; he never commented on political matters or equality.
So, the short and long answer is there is no connection. There is nothing about the ideas of the Great Awakening which emphasize or even discuss political equality; rather the aim of the Awakening was to scare people away from hell. If one considered British rule to equate to the place of eternal punishment, then perhaps there is a connection; otherwise it simply doesn't exist.
The Great Awakening justified the political thought of the Revolution in the minds of many Americans because the Great Awakening was an egalitarian religious movement.
Leaders of the Great Awakening taught that people did not need to have educated, elite clergymen telling them what to do. Instead, they could find God through their own minds and their own emotions. They could get inspiration from men like George Whitefield who had worked in an alehouse rather than getting it from men who had gone to college.
These sorts of ideas justified the ideas of political equality. If all people could find their own ways to God, then surely all people were equal like the Declaration of Independence said they were.