The influence of Braginskii's viscosity tensor on the rotating, self-gravitating modes of anisotropic plasma is investigated within the Chew-Goldberger-Low (CGL) framework of magnetohydrodynamics. Using normal mode analysis, a general dispersion relation is derived and examined for rotation aligned parallel to the magnetic field. The roles of compressional, shear, and drift viscosities are analyzed in shaping transverse propagation modes, as well as their effect on gravitational instability. Analytical and numerical results demonstrate that viscosity introduces significant modifications to the stability thresholds depending on propagation geometry and rotation direction. These results obtained show the relevance in particularly astrophysical systems such as highly magnetic white dwarfs' environment and magnetar's magnetosphere.