If you set foot in a country, they can arrest you for violating their laws. Doesn't matter if you aren't a citizen and live overseas. If you come there, they can arrest you. So let's say you regularly trash Islam and the Ayatollah and are well known for this. Then you travel to Iran. They very well can arrest you for that. They can't do much if you don't go there but if you show up, they can grab you.
Now in terms of if this particular arrest is legit for the American legal system, almost certainly. Doesn't matter that he was living in a foreign country. If he sold something that is illegal to Americans and using American services, he broke American law. Doesn't matter if he wasn't in America at the time, you don't have to be in a country to break their law. Let me give you a couple examples of how one can easily break a country's law from another country:
1) Ordering someone murdered. Let's say you have yourself a little gang with members in a few countries. You don't like someone over in Sweden so you order one of your Swedish members to murder them. That person broke Swedish law, but so did you. Doesn't matter you weren't there, you orchestrated a murder, that's illegal, and if they can get their hands on you you'll stand trial for it (the US would happily extradite you for that).
2) You set up a gun smuggling business for Canadians. You go and buy guns that are legal in the US, but illegal in Canada. You have them smuggled up and warehoused there, and then sell them to Canadians. You've broken Canadian law. Even if you are operating everything out of the US, what you are doing isn't legal in Canada and that's where it is being done. You house the guns in Canada and sell them to Canadians, that makes it a Canadian issue (you'd get extradited for that too).
So if this dude is selling his shit from AWS, to Americans, the courts will have no problems with the claim that American law applies.