We propose a semismooth Newton algorithm to solve the Navier-Stokes equations with a directional do-nothing boundary condition. This boundary condition is defined in terms of a max-function, which is not differentiable in the classical sense. Our approach consists in first discretizing the equations in a suitable finite element space and, subsequently, applying the semismooth Newton method at the finite dimension level, resorting to the Clark differential of a max-operator. Numerical simulations with exact and unknown solutions are presented to show the advantages of this algorithm over the Picard method.