That's way off. While the US started the rock/pop era, the Beatles, Rolling Stones and Who had massive worldwide influence in the 1960s. The only US band of the time that wasn't completely washed away by that tide, in terms of influence, was the Beach Boys. The British created and popularized heavy metal with Led Zeppelin and Black Sabbath. The UK created glam rock in the '70s (Bowie, Roxie Music, T.Rex, etc). The UK punk scene (Clash, Sex Pistols, Buzzcocks, etc) was the seminal scene of the late-70s and turned into the new wave movement that dominated the early 1980s. That UK punk scene, while not the first instances of punk, is still what pretty much everyone thinks of when they think of punk roots. And punk is a massive part of rock/pop.
Grunge was certainly a major movement in the 1990s, but the UK had their own acid house scene at the same time that pushed dance beats into the pop mainstream.
The Beatles didn't have to come to the US to get beyond their early music style. The two were completely unrelated. The Beatles changed musically as they got older (and it's pretty arguable, IMO, whether they ever made a better song or a more impactful song than "She Loves You").
The US has a great pop history (and there are some forgotten bands, by the mainstream, that had massive influence on music, like Velvet Underground, Big Star and the Pixies), but I don't think it's remotely true that all the worldwide impact on musical direction came from the US.
