I mean a truly amoral view on this matter is that the genetic traits behind Caucasians are finally becoming a recessive trait. Something that any geneticist will tell you was probably inevitable in the first place.
That said, I don't think the North American culture is going anywhere. If anything is stronger than ever... It's spent the last century slowly assimilating the previously dominant European cultures. However, I think in the coming century there is a good chance for the Chinese culture to begin replacing the North American culture.
Now what will be really interesting is to see if the North American culture will ever see a resurgence. One of the hallmarks of the surviving "Old World" cultures is their resilience and ability to rebound...