Despite your user name it is you seem to take "a-historical" position. USA is a product of Protestant/British culture. If WASP core disappears USA will not last very long.
Same was with the Roman Republic - it decayed and dissolved when the ethnic Latins who created it lost their demographic strength and cultural identity.
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.