mgh ~~

Please remind that PE=mgh means that the ENERGY NEEDED TO RAISE THE OBJECT, that energy will be stored in PE. E=mgh can be explain in that this is the work done on the gravitational field. Same as W=Fs, F=mg and s=h.
Assume the object with mass m as put on the table. The table is giving the object a normal reaction F=mg upward, or else the table will break and the object will fall down. But remember that if there is a little rocket supporting the object will F=mg, then the object will not fall, keep in rest. While someone give an initial velocity to the object, the object will rise will velocity v, since the net force on the object is : F=-mg(gravitational)+mg(little rocket)=0, by Newton's 3rd law, there is no acceleration, but increasing in PE. PLEASE REMIND THAT NEWTON'S 3RD LAW IS USED, IT STATED THAT THE OBJECT WILL KEEP IN REST OR UNIFORM VELOCITY WHEN THE NET FORCE IS ZERO.

3rd is about reation