Increasing attention has been recently paid to the development of nanocomposite materials for food application as new tool to enhance the mechanical and thermal properties of polymers. In this study, novel chitosan–clay nanocomposite films were produced as carriers for the covalent immobilization of papain, by using a fixed amount of chitosan (1% w/v) and a food-grade activated montmorillonite (Optigel, OPT) or a high-purity unmodified montmorillonite (SMP), in four different weight percentages with respect to chitosan (i.e., 20, 30, 50, 70% w/w). Both nanoclays (OPT and SMP) improved the mechanical properties of the obtained nanocomposites, and the OPT films showed the highest Young modulus and mechanical resistance (σmax). The nanocomposites were used as carriers for the covalent immobilization of papain, which was preliminarily characterized in model wine towards a synthetic substrate, showing the highest efficiency in the release of the reaction product when it was bound on OPT-30 and OPT-50 films. Finally, the latter biocatalyst (papain on OPT-50 film) was applied for the protein stabilization of two different unfined white wines, and it efficiently reduced both the haze potential and the protein content.