Would the Constitution allow John McCain to choose Arnold Schwarzenegger as a running mate?
Obviously the Terminator would not be able to run for President (having been born outside of the U.S.), but how about VP? A McCain-Schwarzenegger ticket would be an interesting one, as it would attract unlikely voters from California (a predominantly democratic state) and would attract moderate and conservative democrats.
No.
The Constitution says that the VP has to be eligible to be president, so no.