Why not? Provided that the EU Presidency (or the Chairmanship of its "European Council") becomes a matter of popular vote, as opposed to being appointed by the heads of state or government of the 27.
I had blogged on this on 11/10/2009, before the first President of the European Council was appointed. And tweeted about it around the same time.
The idea, for those who like to think outside the box and think ahead is not new. I had thrown it in back in August 2004 in the Guardian Forums at the height of the US hype re the 2004 US elections that re-elected GWB.
It was then clear that Mr. Schwarzenegger, a moderate Republican, then governor of California would make a good candidate for the US Presidency, except for the fact that he was not born a US citizen. And that the experience of the Austrian born world class bodybuilder, then actor, then businessman then politician, as the governor of one of the world's biggest economies (California) would be a shame to waste.
It was also clear , not only in 2009 but also back in 2004, that it would be very difficult to find a European of pan-European popular appeal in case the EU Presidency became the subject of a vote by the EU citizens.
Hence the appeal, even far fetched (?) of an Arnold Schwarzenegger candidacy for an upgraded EU Presidency post. Too political sci-fi? Well, IMO only time will tell! Remember this post.