Langmuir monolayers of ligand-capped inorganic nanoparticles exhibit rich morphologies under lateral compression such as wrinkling, folding, and multilayer nucleation. We demonstrate that the ligands play a crucial role in the mechanical properties of nanoparticle films by probing the morphology and anisotropic stress response during lateral compression of films with systematically varied ligand concentrations. Increasing the ligand concentration of the films past a threshold value inhibits monolayer wrinkling and folding in favor of multilayer formation, and sharply reduces the compressive and shear moduli. We attribute these drastic mechanical effects to modifications to the ligand interactions between adjacent particles as well as to two-dimensional crystalline structure changes occurring on the scale of tens of particles, as determined by transmission electron microscopy.