I disagree. I think, of all people, the President needs to have been raised here and had the "American experience" per say. Maybe if he/she wasn't necessarily born here he/she should have been here from a young age.
Wouldn't an "American experience" be coming to this country as an immigrant, and working your way up?
I think if the people of this country actually elect someone who wasn't born in this country, it should be up to the people of this country. It made some sense right after the revolution, when our ties to Britain were still so strong that conceivably we could've elected a British president... who may not have had the interest of the country in mind. As rumborak points out, it's not going to be easy to do, and all you have to do is look at the reaction over the false idea that Obama isn't a US citizen, and the complaints raised about his upbringing, to see what an uphill battle any foreigner would really have.
Though, Arnold probably couldve won. He's a movie star, meaning at least half of America is in love with him.