The Great Awakening contributed to the development of an “independent spirit” among American colonists because it reduced the power of hierarchical religious sects. It contributed, instead, to a situation in which people thought more for themselves about issues of religion.
Prior to the Great Awakening, Americans typically belonged to hierarchical churches. They were told what to believe by their church leaders. They were not really encouraged to think for themselves. We can argue that this would cause them to lack an “independent spirit.”
The Great Awakening reduced the power of such churches. It encouraged people to break away from the belief that worshipping God consisted of obeying certain rules. Instead, it encouraged them to seek personal relationships with God and to seek their own personal understanding of God’s word. This would be likely to cause people to have more of an independent spirit in all aspects of their lives. If they were good enough to be able to think for themselves on something as important as the health of their souls, they would be good enough to think for themselves on issues of government and other such things.
The Great Awakening helped establish an independent American spirit in several ways, but I would like to focus on the way it affected "the individual." The Great Awakening emphasized the role of the individual in salvation and Christian practice. Historically (especially in the Catholic Church), Christians had emphasized the communal vision of salvation--the idea that salvation is found within the Church and that apart from the Church one cannot be saved. Great Awakening preachers taught, in contrast, that one could have a relationship with Jesus Christ independent of the Church.
This elevation of the role of the individual was closely linked to the philosophical ideas of the Enlightenment, which had begun to prevail during this historical era. Many Enlightenment philosophers argued, for instance, that the government derived its powers from the collective group of individual citizens, not from divine or hereditary right. Since every citizen was in a sense sovereign, it only made sense that the form of government should reflect that. Hence the revolutionary spirit promoted a form of democratic republicanism.