> Unless they're also implying that the US government should be allowed to go after UK companies that don't follow it's free speech regulations because American citizens can access them.

Precedent in the US is that courts do in fact have jurisdiction over a foreign website's owner if the owner "purposefully availed itself of the U.S. forum or purposefully directed its activities toward it", a test which is less demanding than it sounds. [1]

And US has taken advantage of this to go after foreign websites such as Megaupload, BTC-e, Liberty Reserve, etc.

Therefore, if there were a US law requiring companies to follow free speech rules, it could potentially be enforced against foreign website owners. But no such law currently exists. The First Amendment itself only applies to the US government (and to companies working on behalf of the US government). There is also the SPEECH Act, which, among other provisions, creates a cause of action where if someone sues a US person in a foreign court over their speech, they can sue back in US court. But only for declaratory judgement, not damages or an injunction. The goal is mainly just to prevent US courts from enforcing judgements from the foreign court in such cases.

[1] https://tlblog.org/how-to-find-personal-jurisdiction-over-fo...