This paper develops and analyzes a fractional-order predator–prey model with Holling type III functional response, incorporating the transmission of a contagious disease between both populations. We first establish the existence, uniqueness, non-negativity, and boundedness of solutions for the fractional-order system. The local stability of the model’s equilibrium points is examined, and the global stability is rigorously proved using a suitable Lyapunov function. We also investigate the effects of disease transmission on the predator–prey dynamics by identifying multiple equilibria, threshold parameters, and stability conditions. In particular, we analyze the existence of Hopf bifurcation at the endemic equilibrium point through bifurcation analysis, revealing the possible emergence of periodic oscillations. Analytical results are complemented by numerical simulations, highlighting the importance of incorporating both Holling type III predation and disease transmission when assessing prey–predator coexistence.