Aristotle didn’t reject the Greek gods, exactly; he could see some social and political utility in maintaining public cults, so he never spoke against them. His view of the universe, however, didn’t have room for them, and he pretty clearly didn’t believe in them. He relied on logic and experience, both individual and collective, as his sources of authority. Since Greek city-states had tried a variety of different political systems, he analyzed them and laid out his opinions for what worked best under what circumstances. He never assumed that the purpose of politics was to serve God and not man, or that the political system of the day was divinely inspired and thus unchangeable.
Notwithstanding the rubbish you frequently read in right-wing history books, the most prominent Founding Fathers were Deists, not Christians. They believed in a creator God who left men pretty much to their own devices. Their views on politics were based on their opinions of human nature, which largely revolved around concepts of class and material self-interest, not on revealed religion. Finally, they relied on the historical record regarding republics, including the Greek and Roman experiences, in the process of drafting the Constitution.
And so, when you read that James Madison ordered as many books as possible regarding the rise and fall of European republics to assist him in putting his ideas together, you are seeing evidence of Aristotle’s DNA in the American system.