We report x-ray resonant magnetic scattering and resonant inelastic x-ray scattering studies of epitaxially strained Sr2IrO4 thin films. The films were grown on SrTiO3 and (LaAlO3)0.3(Sr2AlTaO6)0.7 substrates, under slight tensile and compressive strains, respectively. Although the films develop a magnetic structure reminiscent of bulk Sr2IrO4, the magnetic correlations are extremely anisotropic, with in-plane correlation lengths significantly longer than the out-of-plane correlation lengths. In addition, the compressive (tensile) strain serves to suppress (enhance) the magnetic ordering temperature TN, while raising (lowering) the energy of the zone-boundary magnon. Quantum chemical calculations show that the tuning of magnetic energy scales can be understood in terms of strain-induced changes in bond lengths.