I think the question who's more fundamental Gravity or Quantum mechanics is like the old dilemma between the continuous and the discrete. In this case it seems that whenever we assume one we bump into the other as a limiting case. My modest opinion on this is we need something new; a new concept a generalization that has them both as limits.
There is a cause effect relationship. In Newtonian formulation of gravity, mass of an object acts as a source whereas the gravitational field is the effect. In Einstein's formulation of gravity energy-momentum acts as the source and the effect is the curvature of space-time.
The distinction between classical and quantum cases lies in the associated length scales. If a mechanical problem (translation or rotation of a single particle or many particle system) is defined in the atomic or subatomic scales, then the problem is quantum mechanical, where the location/momentum of such tiny objects with respect to time no longer obeys deterministic laws; and in turn becomes probabilistic.
Both energy and momentum possesses an intrinsic uncertainty in the sub-atomic world. Energy is uncertain for fixed time coordinates, whereas momentum is uncertain for given space coordinates. Now let us apply Einstein's ideas, where in a causal structure source is energy-momentum which is uncertain; this can never lead to an effect devoid of uncertainty; consequently now we can see that space-time in quantum gravity should have an inherent fuzziness.