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.
Are you actually saying that a particular aspect of the Roman economy/society is applicable to something entirely different and completely irrelevant in the modern day?
The European colonies in eastern North America were started by the successor states of the Roman provinces of Gaul, Hispania, Batavia and Britannia. Two of these spoke Latin derived languages, and the other two spoke languages heavily indebted to Romance languages. (Much of English core vocabulary is either French or Latin in origin — 60% according to one estimate.)
Other Roman influences on the USA include Christianity and to a lesser extent Judaism (since the major US versions come out of post-Roman Judaism.)