In this essay, O'Sullivan argues that the most important principle that guides America is the principle of equality. We can see this, among other places, in the following passage.
The American people having derived their origin from many other nations, and the Declaration of National Independence being entirely based on the great principle of human equality...
O'Sullivan is arguing that the fact that the US is based on this principle makes it unique and superior. O'Sullivan argues that other countries have been based on principles that are corrupt and immoral. Because of this, those countries are not great countries like the US is and do not deserve to have a great destiny. He says
What friend of human liberty, civilization, and refinement, can cast his view over the past history of the monarchies and aristocracies of antiquity, and not deplore that they ever existed?
To him, the principle that a country is founded on determines its worthiness. The US was founded on a great principle--that of equality. Therefore, it has a great destiny.