We present a coupled high-fidelity numerical approach for accurately modeling the fully nonlinear wave–structure interaction of a floating highly-flexible structure in waves. The methodology combines numerical methods that solve the Navier–Stokes equations for viscous-flow dynamics with a free surface, and formulates the nonlinear dynamics of the flexible floating structure as a set of differential–algebraic equations. A verification study was conducted for numerical wave generation and propagation, and the discretization error was quantified. The validation was performed against experimental measurements of a flexible structure subjected to regular waves, where its upstream edge was constrained to the vertical equilibrium location. Our comparisons with the experimental data showed a favorable agreement. This study specifically focused on comparing the wave-driven hydroelasticity of a constrained and a freely floating flexible structures. It provided insights into the effects of rigid body motions on its structural deformations in waves. Developed as an open-source tool, our study demonstrates the potential of the coupled methodology for accurately modeling the wave–structure interactions of flexible floating structures in various nonlinear sea states.