It's highly unlikely to happen, it's just too much hassle to change the constitution. Certainly it's not going to happen while one major party has a credible foreign-born contender, the other party would be hell-bent to prevent it.
I guess Arnold would probably win, not that I like the prospect.
Of course no civilized nation would ever ban foreign-born citizens from political office, the only places where I know of such a rule are the US and Cote d'Ivoire, where it was introduced to keep the most popular candidate, Alassane, I've forgotten the surname, from running. (He was born in Burkina Faso.)
In Finland we also have such a rule. President of Finland should be native Finnish.
