Einstein's equivalence principle (EEP) tells us that there is no way in principle to locally distinguish between inertial acceleration and the effects of a gravitational field by carrying out any non-gravitational physics experiments. Furthermore, in a free-falling frame of reference the laws of physics should reduce to those of special relativity (SR).
My question is, can it be inferred from the EEP that the laws of physics must be the identical in all frames of reference, i.e. they must be generally covariant?
Having thought about it, I think it’s the case that although the principle of equivalence and general covariance do not imply one another, it is the case that the principle of equivalence, in particular, the EEP, that gives physical content to the principle of general covariance. The reason being that, by itself, the principle of general covariance is vacuous since any physical theory can be expressed in a general covariant form. What gives it physical content is that the EEP implies that the (non-gravitational) laws of physics reduce to those of special relativity in the presence of an arbitrary gravitational field, so long as one is in a local inertial frame of reference. Given this, we can use the notion of general covariance to re-express the equations in this frame in tensorial form. These equations will then reduce to their special relativistic form whenever one is in a local inertial reference frame. Now that they are in tensorial form, if they hold in one frame of reference, they must hold in all frames of reference (both inertial and non-inertial). This is exactly the requirement of general covariance.