The Gauge-Uzawa method [GUM], which is a projection type algorithm to solve the time depend Navier-Stokes equations, has been constructed in [14] and enhanced in [15, 17] to apply to more complicated problems. Even though GUM possesses many advantages theoretically and numerically, the studies on GUM have been limited on the first order backward Euler scheme except normal mode error estimate in [16]. The goal of this paper is to research the 2nd order GUM. Because the classical 2nd order GUM which is studied in [16] needs rather strong stability condition, we modify GUM to be unconditionally stable method using BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correction method and the errors are already estimated in [8] for the Stokes equations. In this paper, we will evaluate errors of the stabilized GUM for the Navier-Stokes equations. We also prove that the stabilized GUM is an unconditionally stable method for the Navier-Stokes equations. So we conclude that the rotational form of pressure correction method in [8] is also unconditionally stable scheme and that the accuracy results in [8] are valid for the Navier-Stokes equations.