If you are standing on the earth and you drop a ball, the ball falls to the ground due to the force of gravity acting on it. Now imagine you are in an upward accelerating elevator in space. You let go of a ball that you are holding in your hand. The ball Is stationary in space, but the elevator floor is accelerating upward and hits the ball. This means that relative to an observer standing on the floor of the elevator, the ball appears to fall to the ground. The ball falls in the exact same way as it does when you are standing on the earth. According to the principle of equivalence, no experiment from inside the elevator can tell whether the ball is falling or whether the elevator is accelerating upward. Therefore, the experience inside an upward accelerating elevator is indistinguishable from the experience of standing on the earth in the earth's gravity. What's strange here is that no force is acting on the ball in the elevator causing it to fall. However, when standing on the earth, the earth's gravity IS acting on the ball causing it to fall. So, how can a ball fall inside an elevator WITH NO FORCE ACTING ON THE BALL? How is this possible? In one situation (standing on the earth), A FORCE is acting on the ball. In the other situation, NO FORCE is acting on the ball. So even though one situation has a force acting on the ball (standing on the earth), and the other situation has no force acting on the ball (elevator), both situations are an identical experience where the ball falls to the ground. How is this possible?
The insight of Einstein is that gravity is not a force, but, rather like the accelerating elevator, simply a redefinition (locally) of what an inertial reference frame is.
That is Newtons model of gravity. In General Relativity, which is based on the equivalence principle, there is no real force acting on free falling objects in a gravitational field. See this animation for comparison: https://www.youtube.com/watch?v=DdC0QN6f3G4
I may not know Einstein's theory, but, from a Newtonian view, there is no contradiction. If the elevator is accelerating, shouldn't it take a force to have the ball not fall? (Because "not falling" means moving with the elevator, which was said to be accelerating.) Newon's law does not say, "forces act on 'falling' objects," because "falling" can be subjective (as your post illustrates).
To clarify the different explanations: Newtonian Gravity: The frame of the Earth's surface is inertial (ignoring rotation etc.): The ball accelerates down due to the force of gravity The frame of the accelerating box in space is non-inertial: The ball accelerates down due to an inertial force or curvlinear space-time geometry. General Relativity: Both frames are non-inertial: The ball accelerates down due to an inertial force or curvilinear space-time geometry.
In a coordinate independent sense, the ball doesn't fall, it travels in a straight line called a geodesic. The elevator is accelerating due to some force and the elevator's path curves up to intersect the ball's geodesic (straight path).
This does not require GRT, just Newton and Galilei. No force is acting in the rest frame of the ball. In the rest frame of the elevator however a gravity force IS acting.