Other than the senate, eagle statues, architecture, philosophical/intellectual tradition?

Not as much of a parallel as people tend to think, but the Roman empire fell apart partly because it was based on the theory of constant outward expansion. If your model, say economic, is based on infinite expansion in a finite arena you'll have trouble.

Ancient Greece then?

Western Civilization saw its start in Rome, and continued through the rise and fall of the Roman Empire when it contracted. The core of Western Civilization reemerged in England in the 15th century and expanded again, with the core moving to France in the early 19th century, before moving again to Germany in the middle of the 20th century. Post-WWII the core moved once again, this time to the United States, and after the expansion of the US Empire, we now see Western Civilization faltering and contracting under unpopular wars driven by a long series of self-serving poloticians. Whether Western Civilization can reorganize and expand (with the core migrating once again) or wither and die under the expansion and invasion by other civilizations remains the great question of the present day.

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.)