What causes a fuse to blow, the current or the power?
It's the watts dissipated in the fuse itself not the watts in the system. Therefore since the fuse has resistance (R) it's the current, which provides that power I^2*R.
The voltage has nothing to do with it : at 6V, 12V or 240V, the fuse still blows at 20A. However you cannot use a low voltage fuse in high voltage applications : it will still blow at (strictly, slightly above) its rated current, but may sustain an arc that a HV fuse would extinguish.