Wikipedia’s entry for Rugby Union has a “global reach” section which may surprise some readers, especially the entries for the likes of Japan, Argentina and Romania.
But Rugby is still predominantly a “Commonwealth” sport, and only a minor sport in Japan. I think kidchameleon makes a good point.
I take it you didn’t read the Wikipedia entry then.
“Rugby union in Japan is a major sport. Japan has the fourth largest population of rugby union players in the world and the sport has been played there for over a century. There are 125,000 Japanese rugby players, 3631 official rugby clubs, and Japan is ranked 14th in the world”
That’s true, but the point isn’t that everyone plays rugby; nobody claimed this. The point is that pretty much all of the rest of the world’s most popular sports are played at the highest level internationally. It’s irrelevant that rugby and cricket are only played in a subset of countries; the point is that the USA as a country is pretty rare in that its most popular sports have very little international competition worthy of the name.
More adults play Rugby in the USA than in New Zealand, according to Wiki and it’s growing.
The American Eagles may not be household names in the USA, but they are proud of representing their country, non the less.
Here, for example, is Takudzwa Ngwenya absolutely destroying South Africa’s defence for what was easily the try of the 2007 world cup. The guy he leaves for dead is one of the top wingers in the world.
Okay, so Ngwenya’s originally Zimbabwean and plays in France, but never mind. 
Agreed. This was the point I was trying to make earlier.
And is demonstrably false, rugby especially so. See Japan having the fourth largest population of rugby players (according to Wikipedia). Unless I missed the bit where they entered the Commonwealth. Or Argentina, a really quite strong side. Not only are they not in the Commonwealth but the UK went to war with them not long back.
But even granting that only about nine nations play union to a high standard (which is pretty much true), it completely misses the point of the OP’s question, which had nothing whatsoever to do with whether the whole world played a particular sport, but asked whether it mattered that the top level of any given sport was international, or domestic.
Exactly how many sports are we talking about that have the model the US doesn’t adhere to? Please list them.
If it’s just a unique thing to soccer, the US doesn’t miss it because the US doesn’t care about soccer.
I guess the real question is how does the rest of the world feel about playing lame sports.
And again, I’ll ask “What about Canada? Do they get a pass?”
Soccer, rugby and cricket. That’s all I can think of. Even then, there’s a US soccer team that participates in international competition, including the World Cup.
There’s non-Olympic international competition that includes US athletes in individual sports such as track and field, tennis, golf, gymnastics, and so on. American curling teams or “rinks” regularly participate in international bonspiels.
So basically, the answer it that the US does, in fact, already participate in international sports the same way everyone else does. We just don’t care about soccer. Seems like a simple enough answer.
One thing to keep in mind is the idea of “local” teams or playing for national pride. America’s identity is a melting pot. Watch the Olympics and check out various countries’ teams. Most of the participants look pretty emblematic of their country. The Asian teams probably have the characteristic more than European ones, but I would submit that America has the least amount of “pure American” participants, whatever that concept would even mean for Americans. We’re a nation of immigrants, so we’re used to having a bunch of disparate backgrounds joining together to make common cause.
That’s why it’s fair to point to NCAA teams as comparable to regional teams in other countries. Conceptually, going away to college to form your identity as an adult is analogous to immigrating to the US to start a new life. College kids represent their local area in a very real sense regardless of where they spent their childhoods.
Also consider that the US has four wildly popular sporting leagues. (NFL, MLB, NBA, NASCAR.) They also have a half dozen or so popular second tier sports. (UFC, NHL, PGA, ATP/WTA, MLS, MLL, etc…) This doesn’t even count the hugely popular NCAA levels of football and basketball.
Let’s consider the UK to be the mecca of international competition. How many wildly popular sports do you guys have leagues for?
This entire thread is an international competition. And nobody’s winning.
Domestic leagues that are on national television? (Association) Football, rugby union, rugby league and cricket. If we’re counting motor sport, Formula 1 is roughly equally popular to NASCAR in the UK, but it’s an international sport, so it’s hard to compare them directly. UK-only motor racing formulas are significantly less popular than NASCAR.
Now, there are also European football competitions - the UEFA Champions League and Europa League (formerly the UEFA Cup). There are European rugby leagues too.
There are also lower national leagues for the major sports, which act as feeder venues for top-flight teams (and the worst teams in each top-flight league are replaced by the best teams in the second-tier leagues after each season).
UFC, boxing, ATP/WTA tours, PGA and PGA European tours and so on are all just about as popular in the UK as in the US. Golf’s not quite as big; tennis is a bit bigger.
What the heck is MLL?
Major League Lacrosse. Not hugely televised, but very big in the locales where the teams are located.
You’re saying that rugby is as popular in the UK as MLB is in the US? Formula 1 is as popular as NASCAR? Rugby is as popular as the NBA? I don’t believe you.
The various levels are irrelevant; I’m asking how many different sports dominate the sporting scene. The US has varying levels as well, and I would bet dollars to donuts that NCAA football and March Madness are both more popular in the US than minor league soccer/rugby are in the UK.
EDIT: And I’m skeptical that the UFC is as popular in the UK as it is in the US.
Women’s colleges? 
I don’t understand.
I’m basically asking what sports are big in the UK. Are soccer, rugby and cricket all major sports that get continuous daily chatter and draw big ratings? Are there any others that do so?