We consider the Navier-Stokes equations with Navier's slip boundary conditions in a 3D curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the Navier-Stokes equations converges weakly in appropriate function spaces on the closed surface as the width of the thin domain tends to zero. Moreover, we characterize the weak limit as a unique weak solution to limit equations on the closed surface, which are the damped and weighted Navier-Stokes equations on a closed surface.