The magnetic excitation spectrum in the bilayer iridate Sr3Ir2O7 has been investigated using high-resolution resonant inelastic x-ray scattering (RIXS) performed at the iridium L-3 edge and theoretical techniques. A study of the systematic dependence of the RIXS spectrum on the orientation of the wave-vector transfer Q, with respect to the iridium-oxide bilayer, has revealed that the magnon dispersion is comprised of two branches well separated in energy and gapped across the entire Brillouin zone. Our results contrast with those of an earlier study which reported the existence of a single dominant branch. While these earlier results were interpreted as two overlapping modes within a spin-wave model of weakly coupled iridium-oxide planes, our results are more reminiscent of those expected for a system of weakly coupled dimers. In this latter approach, the lower-and higher-energy modes find a natural explanation as those corresponding to transverse and longitudinal fluctuations, respectively. We have therefore developed a bond-operator theory which describes the magnetic dispersion in Sr3Ir2O7 in terms of quantum dimer excitations. In our model, dimerization is produced by the leading Heisenberg exchange J(c), which couples iridium ions in adjacent planes of the bilayer. The Hamiltonian also includes in-plane exchange J, as well as further neighbor couplings and relevant anisotropies. The bond-operator theory provides an excellent account of the dispersion of both modes, while the measured Q dependence of the RIXS intensities is in reasonable qualitative accord with the spin-spin correlation function calculated from the theory. We discuss our results in the context of the quantum criticality of bilayer dimer systems in the presence of anisotropic interactions derived from strong spin-orbit coupling.