Are you asking whether he had the right three steps or are you asking whether the US can/could become "time's noblest empire?" It's not completely clear from your question.
Adams asserted (this was at a grounbreaking ceremony for the Chesapeake and Ohio Canal) that the three necessary steps were:
- Getting the right of self government as the US did when it broke from Great Britain.
- Uniting the states into one country.
- Putting all the efforts and powers of all the people into improving the country. By this, he means improving the country both physically (as with the canal) but also politically, morally, and socially.
It is hard to argue with the idea that these three steps are necessary for any country to become "noble." The first and last steps, in particular, are things that any country must go through in order to be the best country it can possibly be. So, if you are asking if those were the right steps, I would say that they are, although that is partly because the third step is such a broad one.
Can the US be "time's noblest empire?" It seems a little arrogant to assert that we could be. The US is certainly a great country that has afforded its citizens a high degree of liberty and prosperity. However, it certainly has its flaws, among which have been economic inequality, racial discrimination, and a tendency to overreach in foreign affairs. So the US is not perfect and never will be.
Of course, no country can be perfect. While I cannot say the US is the best possible country, neither can I say that there is any country better than it. So it could be that Adams is right that Adams is right and the US is "time's noblest empire," but I don't think we can say objectively that this is so.