Taking a semi-opinionated swag, the core went from Rome to Constantinople, then Venice, then London.
Looking from a Finnish perspective, Sweden and Estonia are culturally closest to us, but Russia is also pretty close. Britain and France already feel less familiar, not to speak of something like Italy or the US.
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.
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.)
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.)
By this logic, Turkey is a continuation of the Roman Empire too. See: https://en.wikipedia.org/wiki/Ottoman_claim_to_Roman_success...
At what point does something stop being a continuation of something else?
Byzantium was certainly a continuation of Rome and its subjects called themselves Romans. It all depends on whether you consider the Ottoman Empire to be a continuation of Byzantium or not (there are arguments both ways).