NO, actually the US and the thought behind it was the product of European culture, but not necessarily WASP culture. Many of the ideas in our founding documents were from French thinkers, the concept of natural law upon which it is based is a Catholic concept that was the foundation of rights and law in Europe even among people who rejected Catholicism, and many of our first settlers were not of English descent.
You may be interested in knowing that one of our biggest problems, slavery, was the result of the English influence. The British (it is thought because of their contact with the Muslims in the Mid East and Africa) practiced chattel slavery in their colonies. The Spanish did not; slavery in Spanish colonies was essentially indentured servitude, with slaves retaining personal rights such as the rights to receive religious instruction, be married, be baptized, and buy their freedom or be freed at a certain point. Many African slaves escaped to Spanish colonies such as Florida and Mexico. The British hated the Spanish for this, pursued the slaves to these areas and tried to punish the Spanish heavily for this. St Augustine, Florida, was burned and destroyed several times by English raiders from North and South Carolina.
Furthermore, to broaden your knowledge, check out the history of Maryland and also examine the early Spanish colonization of California, where the Spanish attempted to settle the Indians and not exterminate them.
The structure of our entire country and our entire history is not based on being a WASP, and is also different from that of the Romans, btw. Anybody can become an American by accepting our ideas. The problem is not that immigrants of any type don't want to accept them, but that the intellectuals of this country have rejected them and are not promoting them. Also, the other problem is that many people appear to be ignorant of our history and go for a strange WASP-supremacist version, which automatically excludes anybody else.
Yet the fact of British settlers importing some French or Catholic ideas is VERY different from them BECOMING French. Otherwise New England would the extension of Quebec.
The structure of our entire country [...] is also different from that of the Romans, btw. Anybody can become an American by accepting our ideas.
What do you mean by acceptance of ideas? When Germanic immigrants accepted Roman dress, titles and citizenship, said and thought that they "accept Roman ideas", and styled themselves as Romans, did they become Roman? It is what everybody liked to think at that time. Yet when the ethnic/cultural core or Latin weakened Rome changed meaning.
Despite that the late Roman experiment of melting pot and absorption of mass immigration and of other ethnic/cultural lands WAS MORE SUCCESSFULL than we tend to think about. even with the Latin core being marginalized.
The FORGOTTEN and quite successful experiment was the Eastern Rome - Byzantine Empire which combined Roman statecraft/statehood with Greek culture and Eastern spirituality (Christianity). It lasted THOUSAND years.
The ethnic core group which replaced Latin place of leadership - the Greeks, accepted Roman identity and ideas of state/government. Latin was the official language for a long time in the New Rome by the Bosphorus, the Roman Law (perfected later by Emperor Justinian), Roman institutions and titles were preserved.
But there are two key elements that might be missing in other attempts to emulate Byzantine experiment. First Greeks the people among whom it was implemented were the most talented, creative, lively and curious that walked on the surface of the earth. Second the Roman state in its original Latin form was already imbued with Greek ideas making it easier to perform this transplant.
And what is telling after Eastern Rome was destroyed by the Turks and after centuries of Muslim yoke the people who emerged from ruins are and consider themselves Greek! (Even if they retain some elements of adopted Roman identity.)