The best explanation of the impact of Greek and Roman democracies on how the U.S. legislative, judiciary, and executive branches function is that while the writers of the U.S. Constitution were familiar with Greek and Roman political ideas, they avoided implementing these into their government in an identical manner.
This recognizes that while the framers of the Constitution were inspired by concepts from ancient democracies, they tailored those ideas to fit the unique needs and context of the United States, leading to a distinct structure that includes checks and balances among the three branches of government.