We investigate the impact of modified gravity on the stochasticgravitational-wave background (SGWB) generated by a cosmological population of unresolved binary black hole mergers. We consider two complementary classes of beyond-General Relativity (GR) effects: waveform-generation modifications described within the parametrized post-Einsteinian (ppE) framework and cosmological propagation effects associated with a modified gravitational-wave luminosity distance. Astrophysical uncertainties in the binary black hole population are consistently incorporated using a Power-Law plus Peak mass model combined with a Madau–Dickinson merger-rate evolution. Using SGWB forecasts for Advanced LIGO at O5 design sensitivity, the Einstein Telescope (ET), Cosmic Explorer (CE), , we perform injection-recovery analyses jointly varying modified-gravity and astrophysical population parameters, including a joint injection in which waveform-generation and propagation deviations are present simultaneously. We show that frequency-dependent ppE corrections produce characteristic distortions in the SGWB spectral shape and can be meaningfully constrained by third-generation detectors, with the CE+ET network providing the tightest constraints. In contrast, modified propagation effects mainly induce smooth amplitude rescalings, are recovered with systematically lower accuracy, and exhibit stronger degeneracies with the merger-rate evolution. In the joint injection, the spectral-shape information is preserved while the amplitude-like parameters retain a residual mutual degeneracy. Our results demonstrate that future SGWB observations will provide a complementary probe of gravitational physics across cosmic history and may open new avenues for testing deviations from GR beyond individually resolved compact-binary events.