Galactic cosmic rays (CRs) inside the heliosphere are affected by solar modulation. To investigate this phenomenon and its underlying physical mechanisms, we have performed a data-driven analysis of the temporal dependence of the CR proton flux over the solar cycle. The modulation effect was modeled by means of stochastic simulations of cosmic particles in the heliosphere. The model was constrained using measurements of CR protons made by AMS-02 and PAMELA experiments on a monthly basis from 2006 to 2017. With a global statistical analysis of these data, we have determined the key model parameters governing CR diffusion, its dependence on the particle rigidity, and its evolution over the solar cycle. Our results span over epochs of solar minimum and solar maximum, as well as epochs with magnetic reversal and opposite polarities. Along with the evolution of the CR transport parameters, we study their relationship with solar activity proxies and interplanetary parameters. We find that the rigidity dependence of the parallel mean free path of CR diffusion shows a remarkable time dependence, indicating a long-term variability in the interplanetary turbulence that interchanges across different regimes over the solar cycle. The evolution of the diffusion parameters shows a delayed correlation with solar activity proxies, reflecting the dynamics of the heliospheric plasma, and distinct dependencies for opposite states of magnetic polarity, reflecting the influence of charge-sign-dependent drift in the CR modulation.