Good question. I'd say it the other way: A good theory of the gravitational interaction should obey (at least the weak) equivalence principle since it's empirically so well established.
As demonstrated by this discussion the problem with the equivalence principle or rather the equivalence principles is that it is usually discussed in the heuristic introductions to GR and then not further qualified given the fully exposed theory (a fate it shares with the discussions of the Michelson Morley experiment, which is usually also only discussed as a "crucial experiment" for the heuristical motivation of SR).
A clear way to state the principle is that the GR spacetime model is a Pseudo-Riemannian manifold with the fundamental form of the signature (1,3) or, equivalently (3,1). Physically that means that the notion of inertial frames is local an that at any point in spacetime there's always a local inertial frame of reference. That's of course not a good way to heuristically motivate GR but it should be the final clarifying statement about the content of the (then even strong!) equivalence principle.
For particle physicists another convincing semi-heuristic argument is that relativistic models of the gravitational interaction can be built by "gauging" Poincare invariance, which becomes then a local symmetry and thus you have a gauge theory with the general diffeomorphism invariance as the gauge group, which usually is called "general covariance". Together with the fact that there are particles with spin that leads to the statement that GR spacetime is described as a Einstein-Cartan manifold with torsion. In the usual "macroscopic" phenomenology, where gravitation plays a practical role, all you have is classical matter ("continuum mechanics") and the em. field as sources, and there the theory specializes to usual GR. If there is torsion, it's inside "polarized matter" and thus pretty difficult to observe.