Single-crystal elastic moduli Cij, shear modulus G, and Zener anisotropic ratio A of solid argon having the face-centered cubic (fcc) structure were determined at high pressures between 12 and 64 GPa using a combination of experimental and theoretical approaches. The experimental data, namely, the maximal and minimal values of the product of the longitudinal sound velocity and refractive index n·VL, were obtained from the 3D scanning of elastic inhomogeneities in compressed samples of argon using the technique of time-domain Brillouin scattering. These inhomogeneities, caused by elastic anisotropy of solid argon, were revealed to be about twice as strong as those reported in the earlier experiments using classical Brillouin light scattering (BLS). To derive the VL values, we used the refractive index obtained here from ab initio calculations which also permitted us to rule out any contribution to the amplitude of the observed elastic inhomogeneities of the hexagonal close-packed phase of argon, proposed to coexist with the fcc phase at high pressures. From the measured Ci j(P), we derived pressure dependence of shear modulus of the fcc argon GH (P) using the Voigt-Reuss-Hill approximation and found a very good agreement with the earlier G(P) obtained from shear sound velocities in the classical BLS measurements. Our results agree very well with the earlier predictions based on a relatively simple many-body model employing the Buckingham pair potential. Finally, our measurements show a much weaker change of the Cauchy discrepancy (C12 − C44 − 2P) of the fcc argon with pressure than reported in all earlier experimental and theoretical works.