They can stop the distribution in the United Stares. In addition, Swedish courts would probably honor the decision of the US court.
Restitution could come from revenues of any US product sales or assets, not just the specific one being contested.
Anything else is an international thing, and unless covered by treaties signed into law between countries, courts from one country never just honour some other countrie's legal system.
Extradition is possible on crimes where both countries agree it's a crime and generally have an extradition treaty, or policy for such requests, but to think that the US courts are somehow global is factually incorrect.
edit: ok, any app store downloads probably could be shut down.