But if it was an American, residing on European soil, there would be extradition procedures to follow. And those would involve having the local (EU) police generate their own warrant and make their own arrest based upon a formal request.
If you had followed this case, you would know that this is exactly what the US tried to do.
The US asked an Irish court to issue a warrant to force production of the data. The Irish court refused to issue the warrant. So, the US issued a subpoena to Microsoft, who rightly told the US that although the data was on a Microsoft computer, the data was owned by a customer of Microsoft, therefore a warrant would be required. The US court then issued a warrant for Microsoft to produce the data. Microsoft refused, noting that the data was in a foreign country, and warrants are only valid when issued by a court that has jurisdiction over the location of the requested object/data/person. No US court has jurisdiction over Irish soil, thus we end up at today's story.
The actual point of Microsoft's appeal is that the US wants to have a court to be able to issue an order that has the all the advantages of both a warrant and a subpoena, while ignoring their limitatations. The problem with this is that subpoenas are allowed to be fairly vague and apply to anything that is "owned" by the target of the subpoena, regardless of where it is located. Warrants, OTOH, can force the target to hand over something they don't own but over which they have control, but can only request very specific items/data, and have to be issued by a court that has jurisdiction over where the item/data is located.