This is a really interesting thread because I've wondered what the US would do too. I just remember thinking after the Beijing Olympics how they had soooo much culture, thousands of years...the US is just a "baby"!
Thanks, WDWgirl03, for taking the thread back to where it started out. (We seem to have three or four different debates going on in this thread.)
"What would American culture look like?" N.Bailey had a great list, and I love the UK thread that Bavaria posted the link to.
When you start thinking about what could be shown in an opening ceremony in the U.S. that would compare to that in Vancouver, there's just so much to work with.
You could start with history. Without trying to copy the show in the American Pavilion in EPCOT, you would definitely feature the Native Americans, the first settlers, the Founding Fathers, the entire "Westward Ho" bit, the Civil War, the waves of immigrants, all the way up to the Space Race. If you want to get in a little grit and not just have a "feel good" performance, bring in references to slavery, the conflicts between the Native Americans and the settlers, the poverty in inner cities, the Depression, the Dust Bowl, the race riots during the 1960s, Vietnam and protestors.
You could also do a lot of visuals with the "Melting Pot" idea, as previous posters suggested.
Since an opening ceremony involves people swirling around, you could easily work in something themed on transport. On water, you would have canoes, the first ships carrying the explorers and early settlers, riverboats and rafts, and move on to more modern vessels, from jetskis to catamarans to tugboats to battleships. On land, it would be horses, stagecoaches, covered wagons, the iron horse, and the Model T, followed by a quick run of different cars (an Edsel, anyone?) You could throw in a stampede, but probably those responsible for security (and clean-up!) would nix this idea.
With those cars from different eras, you could also show how fashions have changed, and how the U.S has contributed to fashion. Add in some music from the same time period to set the mood, and for the 1920s you would have flappers (and bootlegging gangsters?) with jazz playing in the background, through to cruising teenagers á la American Graffiti, and up through to the present. American popular music combined with cars - you'll find traces of these types of Americana everywhere in the world.
And yes, Hollywood - in the sense of both movies and television - would have to feature in there somewhere, perhaps the one American "invention" that has had the greatest impact on culture everywhere in the world. Marilyn Monroe and a bevy of starlets; John Wayne; Cary Grant ... the list is endless. (There might be copyright issues in getting Mickey Mouse featured here, but I'm sure our lawyers could work it out.) The entire Hollywood bit could be done in the form of a parade.
Baseball, football, ice hockey, extreme sports? Get a few designers together, and they could easily convey the grace and beauty of those sports, with the counterpoint of raucous announcers and fans. Throw in a few coca-cola sipping, hot dog -munching fans for humor.
So far, I don't think anyone has mentioned one particular aspect of American culture, inventiveness. Part of our melting pot tradition is that we have attracted the best and the brightest, and the result is all kinds of inventions - America's gifts to the world - that could be featured. Telephones, airplanes, television, the entire idea of the assembly line, all kinds of inventions in electronics, medicine, transport, whatever - all of this definitely counts as American culture. (Yes, yes, someone outside of the U.S. may have been first off the bat with early forms of these different inventions, but there are so many that really saw the light of day here.)