Not necessarily, or if it does, it'll take a very long time. Remember, the US states' cultures were all mostly from Britain, and everyone spoke English with a few exceptions (like the Pennsylvania Dutch). Early on, there were settlements by the French, Dutch, Spanish, etc, but the British settlers pushed everyone out (the French only survived in Quebec, which isn't part of the US).
Wow, this is sooo wrong. Just about the only commonality that the U.S. population started out with was that we are all, every single one (including American Indians and Eskimos), immigrants from somewhere else. The U.S. has seen waves of immigration from all over the world.
As a personal example, I'll cite my great-grandfather. He helped settle Chisholm, a small town in northern Minnesota in the first decade of the last century. He was a Serbian peasant whose family had spent about 250 years in Croatia but still considered themselves Serb, not Croatian. Still used the Cyrillic alphabet attended the Serbian Orthodox services at somebody's house rather than attend the local Catholic church. Then he gets to the U.S. and everything changed for him.
His new neighbors were Welsh, Italian, Jewish, Slovenian, Russian, German, Norwegian, Finnish, and FSM knows what else. All of those families were founded by peasants right off the boat who had come to work in the iron mines or in the logging industry.
The Welsh were coal miners who got jobs as mine foremen because they were typically the only ones underground who spoke English, which in turn meant that they were the only ones who could talk to the mine management. The rest just showed up at the mine for their shift and got by with a lot of hand waving.
Eventually, they all learned English, took night classes to earn their citizenships, made sure their kids were brought up speaking English, and generally became members of the American culture. But every last one of those families is still fiercely proud of their own distinct heritage and celebrates their differences as well as our shared commonalities.
In the past several decades, Minnesota has seen large influxes of Hmong, Vietnamese, Somali, Afghani, and a couple of other refugee groups. We've even got Mexicans who have chosen to settle here instead of following the crops. Those families have all followed similar paths. We've got a huge Cinco de Mayo celebration in the state capital every year.
(As an aside, why on earth are so many people from the tropics so happy to move to the nation's icebox? :-D)
(As another aside, the far right's screaming about illegal immigration is one of the dumber things that I've ever seen in my life. After all, compared to the Indians and Eskimos we're all newbies.)
The point to remember is that America has never really been a melting pot. We're more of a stew, where each new immigrant population adds its own distinctive flavor.
When I look at the history of Europe since about 1970, I see the same thing happening. It's slower because the national boundaries tend to contain each distinctive national flavor, but trust me. There is already far more commonality across Europe today than there was 40 years ago. It may be hard to see from the inside, but it's there.