If gravity is in reality spacetime geometry why when I drop an object on the surface of the Earth does it fall to the ground? Does spacetime push it?
What is exactly your thinking behind "gravity is space-time geometry"?
In layman's terms, general relativity tells us that a particle with some defined mass will curve space-time, causing a gravitational attraction to be felt by the objects around it. When you drop an object on the surface of the Earth, the curvature of space-time caused by the large mass of the Earth, relative to the small mass of the object, causes it to be attracted by gravity and fall to the ground.