What are the chances the United States (and others) follow suit? For startups, I assume most will continue to live on the App Store lest they force their users into two different experiences based on their location.
But it depends how this plays out in Europe. If 95% of popular apps stay on the App Store, then there will continue to be no appetite for change in the US.
But if suddenly Europe is getting different, better apps than Americans are, there might be popular demand to change things. But in that case it's still easier to see Apple making the changes out of public demand, rather than US legislation.
The most likely change in the US I would expect is Apple deciding to let Chrome and Firefox distribute their own browser engines in the US, if Chrome with a Chrome engine becomes popular in Europe and provides a lot of features Safari doesn't.