We first analyze a stabilized finite volume method for the three-dimensional stationary Navier-Stokes equations. This method is based on local polynomial pressure projection using low order elements that do not satisfy the inf-sup condition. Then we derive a general superconvergent result for the stabilized finite volume approximation of the stationary Navier-Stokes equations by using a $L^2$-projection. The method is a postprocessing procedure that constructs a new approximation by using the method of least squares. The superconvergent results have three prominent features. First, they are established for any quasi-uniform mesh. Second, they are derived on the basis of the domain and the solution for the stationary Navier-Stokes problem by solving sparse, symmetric positive definite systems of linear algebraic equations. Third, they are obtained for the finite elements that fail to satisfy the inf-sup condition for incompressible flow. Therefore, this method presented here is of practical importance in scientific computation.