Gravitation - Pulling or Pushing force? According to Newton, gravity is the pulling - or in fact the attracting - force of any heavenly body towards any object to its center. But, to the contrary, Einstein once said that the four dimensions of space and time push the object downwards.

So can which one of them is correct?
I'm guessing that this misunderstanding is a result of the oft-used rubber sheet analogy. The rubber sheet analogy says that, according to general relativity, mass curves space-time like a heavy bowling ball on a near-taut blanket (or rubber sheet) curves the blanket/sheet. This resulting curve makes other bits of matter/energy move in different ways. I'm guessing that this is your confusion.

The rubber sheet analogy fails massively in one area, any demonstration of it involves gravity on Earth. If I use a bowling ball to deform a sheet, and then role a golf ball along the sheet nearby, the golf ball will move a bit towards the bowling ball because of the force of gravity around me - not "gravity" in the simulation. It thus makes it seem like gravity pulls the golf ball "down" because the bowling ball pulls the rubber sheet "down". This is the result of using a two-dimensional analogy of a three-dimensional universe.

The point is, there is no "pushing" going on in the general relativistic model of gravity. Gravity is attractive (so long as the strong energy conditions holds for the object in question), just like Newton postulated.