We describe a model for solid objects surrounded by fluid that accounts for the possibility of acoustic pressures giving rise to damage on the surface of the solid. The propagation of an acoustic pressure in the fluid domain is modeled by the wave equation and the response of the solid is modeled by elastodynamics coupled with gradient damage. The interaction between the acoustic pressure and the deformation of the solid are represented by transmission conditions at the fluid-solid interface. A cohesive phase field method is adopted to represent the initiation and evolution of the damage field. Results from model-based simulations are provided for a benchmark problem and recent experiments in nano-pulse lithotripsy. A parametric study is performed to illustrate how damage develops in response to both the driving force (magnitude and location of the acoustic source) and the fracture resistance of the solid. The results are shown to be qualitatively consistent with experimental observations for the location and size of the damage fields on the solid surface. A study of limiting cases also suggests that both the threshold for damage and the critical fracture energy are important to consider in order to capture the transition from damage localization to fracture.