We discuss the reconstruction of generic 3+1-dimensional space-time geometries from covariant quantum spaces as backgrounds in the IKKT matrix model. Generic classical geometries can be realized within the weak gravity regime, without inducing significant higher-spin contributions. In the strong gravity regime i.e. for strong curvature, the background acquires significant higher spin contributions, so that the classical geometry is no longer adequate. Assuming that the scale of noncommutativity is given by the Planck scale, the weak gravity regime is bounded by a curvature scale of the order $10^{-4} m$, and easily compatible with known gravitational physics. This justifies the framework for emergent gravity given by the semi-classical matrix model, supplemented by an induced Einstein-Hilbert action which arises in the presence of fuzzy extra dimensions.