Here is just a small remark. It is possible to give a strict mathematical proof about the equivalence of these two pictures.
If you just start with the three (semi-experimental) facts: Lorentz invariance, $1/r$ long-range tail of gravitational force and its one-way action (attraction only) and the fact that the bending of light almost doesn't depend on its frequency and polarization, then you will find that these facts are compatible (in the large distance limit) only with the massless helicity $\pm2$ particle exchange. After that, it has been proved that special relativity and analytic properties of scattering amplitude lead to the equivalence principle [1,2]. This theorem is a pure analog of Gell-Mann-Low-Goldberger soft photon theorem, which claims that the power expansion of the amplitude of photon scattering by a hadron (with respect to photon frequency) does not depend on the spin or internal structure of the hadron (up to the second order). By considering multigraviton scattering amplitudes one can prove that the all local vertices for soft gravitons correspond to the expansion of the Einstein action.
It means that the exchange of helicity $\pm2$ massless particle unavoidably leads to the classical general relativity (the opposite statement is trivial).
This program was initiated by Steven Weinberg [1,2] and finished by Deser and Boulware . You can find the complete consideration in their paper  with the title “Classical general relativity derived from quantum gravity”. This paper is a real masterpiece of clear physical explanation of this problem.
 S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. B135 (1964) 1049.
 S. Weinberg, Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations, Phys. Rev. B138 (1965) 988.
 D. G. Boulware, S. Deser, Classical general relativity derived from quantum gravity, Ann. Phys. 89 (1975) 193.