Fractional-order models provide an effective framework for studying epidemiological processes with memory effects. In this study, a nonlinear fractional-order SEIHR model for pneumonia transmission is proposed using the Caputo derivative. The model is analyzed within the framework of nonlinear functional analysis, where the system is represented by a nonlinear operator on a suitable Banach space. Fundamental qualitative properties, including positivity and boundedness of solutions, are rigorously established. Disease-free and endemic equilibrium points are derived, and the basic reproduction number is obtained via the next-generation operator approach. Local and global stability of equilibrium are investigated using fractional-order spectral conditions and Lyapunov functions. Numerical simulations based on the fractional Adams–Bashforth–Moulton method support the theoretical results and illustrate the influence of memory effects on pneumonia transmission dynamics.