In [Nonlinearity, 11 (1998), pp. 1625-1636], Clopeau, Mikelic, and Robert studied the inviscid limit of the two-dimensional incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations, and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include pth power integrable initial vorticities, with p > 2.36411301141