In general, we shouldn't try to compare different historical events or states over such a time period. Because of how much changes and how different the time periods in which people live, comparisons like this tend to wash over myriad differences and present a too-simple picture. Between the US and the Roman Republic/Empire, there are so many ways politically, religiously, societally, economically, etc. etc. that the comparison falls apart.

But that's not really what you're here for, so let's get into it. The way I read this, the comparison you are making is as follows: We have a hegemonic power that spread its culture and eventually declined as a power to be supplanted by rising power that it influenced significantly. Let's go ahead and take that point by point. I'm better versed in the Greek/Roman history, so that's mostly where I will be focusing here.

#1 Was a Hegemonic Power

So, a big note here is that we tend to project our understanding of the world backwards onto history. This is generally unfair and clashes with how those living in history understood things themselves. That gets to the first major failing of this comparison: Britain, at least as far as the mother country was concerned, was a unified state, while Greece was very much not. Thucydides, in his History of the Peloponnesian War, talks about a number of city states in coalition with the Athenians and Spartans during their war. These relationships were not dominant, that is to say, while Athens was the head of its coalition, it was not fully in control of the other members. Throughout the Classical period, Greece, as we understand it, was riven by wars between the city states. All of this is to say that no 'Greek' identity really existed. Really, the first time you can say that 'Greece' as such is unified is under the Romans.

Moreover, I find it a stretch to call Greece a hegemonic power. Certainly, they were important and established colonies all throughout the Mediterranean, but realistically, they did not exercise power over any of the major states of the day. Their colonies operated largely independently of their mother states. If you are looking for a hegemonic power in this period, I think it has to be the Persian Empire - a large, powerful empire that extended and exercised its power on the states around it, including the Greek city states. Of course, that all changes with Alexander the Great and the successor states.

#2 Spread its Culture

I won't dwell too much on this, since it is relatively well understood. Alexander the Great is largely credited with Hellenization, but most of that activity was in the direction of the former Persian empire. Let's just focus in on the influence of Greek culture on the Romans. It is well attested that the Romans loved Greek culture. Most Roman aristocrats spoke Greek. The Roman pantheon is later ages largely took its essential structure from the Roman. Roman literature really only comes out of contact with the Greeks. Horace said that "Captive Greece captured, in turn, her uncivilised / Conquerors, and brought the arts to rustic Latium." (Source). This, however, gets to an important point and the place where this particular comparison falls apart: It was only after conquest that we see a major Greek influence on the Romans. In contrast, the colonies that became the United States were just that colonies. Rather than being separate and influenced by the culture of another power, they were very much a part of that power. That is to say, the culture of the United States wasn't so much influenced by British culture as it was British culture prior to natural divergence. I don't think just cultural spread is important, so much as the manner of transmission. In that, the two just don't compare.

#3 Declined to be Supplanted by a Rising Power

This is arguably the trickiest one. Naturally, there is decline in both the 'Greeks' and the British Empire. But when we think of Greek decline, well, that's really tricky. When did Greece decline? Was it the aftermath of the Peloponnesian War? Was it after Alexander's death? The death of Pyrrhus? Naturally, measuring decline of a classical civilization is really hard! It's easier with Britain, which can be pretty naturally dated to post WWII, when the empire broke up. But for Greece, I think this is much harder.

But let's take the second part of this assertion, which is easier. After all, at some point, Rome did supplant any hegemonic power in the Mediterranean. But here, again, the relationship between Greece and Rome is so vastly different from that of the United States and Britain. Greece and Rome were antagonists and eventually conqueror and conquered. By contrast, the US and Britain were colony/mother country, antagonists, and allies. At the time of Britain's relinquishing of its hegemonic power, it was passing to a friendly nation, whereas the Greeks were more swept up in the rising of a power that they then subsequently influenced. In this too, I just don't think the comparison holds up.

Overall, it is very hard to make effective comparisons. At a 10,000 foot level, I can see how this comparison makes sense, but diving into the details of it, I just don't think it works in a meaningful way.