The way I see it, there are two approaches:
1) If we accept that the sheer mass of the Earth is not curving the surrounding space-time, but, instead, it is caused by us being in a frame of reference that is accelerating to an inertial reference frame, and, the acceleration of free fall is perpendicular to the surface of the Earth at any point, then, we must logically conclude that the surface of the Earth is flat.
2) The other one is if we assume that the Earth is a disk, we have 2 choices:
Either the mass of the earth is curving spacetime and creating an approximately homogenous gravitational field, or the gravitational field is caused by the aforementioned acceleration.