re: Which ancient civilization is the most similar to America?

Roman Republic/Empire

Two executives, bicameral legislature, civil and criminal courts, appellate systems, vetoes, trial by jury, presumption of innocence in court, melting pot, multilingual, multi-cultural, multiple religions, citizen-soldiers, professional athletes, international trade, dominant military power of the age, relatively literate population, welfare system for the poor, continuous war in foreign lands, rose up against foreign power to establish nation, slavery (at one time in the U.S., of course), immigration problems, massive deficit spending... the list goes on and on...

It can be argued that the United States government at its founding was heavily influenced by the structure of the Roman Republic. It's almost a carbon copy.

Definitely Rome.