The parametric high-fidelity generalized method of cells (PHFGMC) is further developed for the prediction of damage and progressive failure of composites undergoing large deformation. The mechanical behavior of the constituents is described using hyperelastic strain energy functions with embedded energy limiters in order to bound the amount of accumulated energy during deformation. The proposed nonlinear micromechanical theory generates both the local and overall global responses, including the effective instantaneous stiffness tensor of the composite. To solve the PHFGMC equations, i.e., equilibrium, continuity and periodicity, and constitutive relations, an incremental-iterative scheme is offered in this study. The iterative scheme is needed in order to develop a progressive failure methodology which enables the determination of the current stress distribution during the applied loading. Applications are given for several case studies demonstrating the effects of the mechanical properties of the constituents (elastic anisotropy and failure parameters), and the geometrical features of the composite (fiber–matrix interphase and fiber waviness), on its overall stiffness and strength. Experimental testing and results from this study and the literature are used for the calibration of the constitutive material properties. This calibration is extensively elaborated and performed to generate the PHFGMC multi-axial effective strength envelopes of the calibrated IM7/977-3 carbon/epoxy composite.