The USA is the product of the British empire which consciously modelled itself on Rome. It was influenced by various European countries, including England and France, which mostly developed in the Roman Empire's former territories.

The European Renaissance and the Enlightenment both used Roman and Greek ideas as inspiration. You can see this in early US architecture, and the models for some institutions such as the Senate. Some of this was ironically as a revolt against English institutions (even though England and Wales were successors of a Roman province).

The early USA's language(s) and Judaeo-Christian heritage are partly products of the Roman Empire. (Spanish most definitely is, and French is. English is heavily influenced by French and Latin.)