A fully nonlinear aeroelastic formulation of the direct Eulerian-Lagrangian computational scheme is presented in which both structural and aerodynamic nonlinearities are treated without approximations. The method is direct in the sense that the calculations are done at the finite element level, both in the fluid and structural domains, and the fluid-structure system is time-marched as a single dynamic system using a multistage Runge-Kutta scheme. The exact nonlinear boundary condition at the fluid-structure boundary is satisfied based on the actual deformation of the wing. The generalized forces associated with the in-plane and out-of-plane degrees of freedom are calculated in local Lagrangian element coordinate systems that fully account for large rigid-body translations and rotations. Finite rotation relations are used to update the nodal deformation vectors at the end of each time step. Numerical results are presented for several nonlinear static and dynamic examples for which published results are available. Results of aeroelastic calculations using the new nonlinear model demonstrate the importance of including the nonlinear stiffening arising from the in-plane strains when calculating limit-cycle-oscillation amplitudes of wings of low-to-moderate aspect ratios and the limitations of the von Karman nonlinear plate model in these cases.