We investigate the stability and convergence of the divergence-free wavelet-based projection method applied to the Navier-Stokes equations. Our analysis shows that the convergence rates of velocity and pressure in the time discretization are of the same order, even when using a second order scheme. This is in contrast to conventional methods, which typically suffer a loss of a factor of 1/2 in the pressure approximation. Numerical experiments on benchmark flows validate the theoretical results and highlight the efficiency of our approach.