Scott already chimed in on that one
We experience gravity through all kinds of things, air pressure is one of the less obvious and immediate ones.
If an apple falls down that isn't air pressure. If a sack of potatoes is heavy that's not air pressure either. Air pressure exists because of gravity, you can't explain gravitational effects by means of air pressure no matter how hard you try so we don't experience gravity as air pressure.Not to mention that air pressure is omni-directional.
We experience gravity through all kinds of things, air pressure is one of the less obvious and immediate ones.