If an object of mass m is on the surface of earth of mass M and radius R then by the law of universal gravitation the force of gravity on the object is given by
F_g= (GmM)/R^2......[1],
where G is the universal gravitational constant.
Again acceleration due to gravity (g) is the acceleration with which any object freely falls under the force of gravity. towards the center of the earth.
Now if the mass of the body be m then by Newton's laws of motion the gravitational pull on the object will be given by
F_g="mass" xx "acceleration"=mxxg.......[2]
So origin of this equation is Newton's laws of motion. Here only acceleration (g) is originated from gravitational force.
Comparing equation [1] and [2] we can write
mg= (GmM)/R^2
=>g= (GM)/R^2
This equation shows how the acceleration due to gravity (g) is related with universal gravitational constant (G) as well as mass (M) and radius (R)of the earth