High-pressure Brillouin spectroscopy of polycrystalline argon, measured using two scattering angles ($180\ifmmode^\circ\else\textdegree\fi{}$ and $70\ifmmode^\circ\else\textdegree\fi{}$), determines the isotropic elastic moduli, shear strength, equation of state, and refractive index of face-centered-cubic argon from 1.3 to 30 GPa at room temperature. The index of refraction $n=1.33--1.67$ over this pressure range. An Eulerian finite-strain analysis (Birch-Murnaghan equation of state) yields an isothermal bulk modulus and pressure derivative ${K}_{T}=15.1(\ifmmode\pm\else\textpm\fi{}1.1)\text{ }\text{GPa}$ and ${K}_{T}^{\ensuremath{'}}=5.4(\ifmmode\pm\else\textpm\fi{}0.3)$ at 2 GPa. The resulting equation of state agrees well with previous x-ray diffraction measurements, illustrating the suitability of high-pressure Brillouin scattering for characterizing the elasticity and strength of polycrystalline materials.