in Newtonian physics only a force can cause acceleration

Something I learnt in 8th grade - if an object is free falling from sky of mass "m" the force it applies when it hits on the ground is m * g, where g is the gravitation.

I think you are mashing up different lessons:

- when you stand on the ground you are the subject of 2 forces: gravitational force m * g and the opposite reaction force of the ground surface, they cancel each other so you stand still

- when you free fall your acceleration is g and the gravitational force is m * g, this is according to Newton's 2nd law

when you hit the ground depending on your velocity and... deformation, the force of impact will be different because you will de-accelerate to 0 or even bounce back very quickly much faster than g. It's gonna be much stronger than g * m

g is not gravity, it’s a constant that quantifies the force that gravity impresses on a mass in Newtonian mechanics

https://en.wikipedia.org/wiki/Gravitational_acceleration Isn't "G" (uppercase G) is gravitational constant and Lower case "g" is acceleration?

yes but they have nothing to do with your assessment "gravity is acceleration". "Forces cause acceleration" is more accurate, there are 4 kinds of forces so gravity is just one cause.