A generalized hydrodynamic model is used to analyse the growth rate of the Jeans instability of a partially ionized strongly coupled plasma incorporating the effects of rotation and Hall current. The general dispersion relation is determined for the propagation of magnetohydrodynamic waves using the normal mode analysis theory. The general dispersion relation is further discussed in four different combinations of rotation and propagation of the system to signify the importance of rotation and neutral particles on the growth rates and conditions of Jeans instability in hydrodynamic and kinetic regimes. The different types of waves are also described in these cases. The influence of rotation and neutral particles on growth rate of the Jeans instability is analysed numerically and shown graphically. The possible applications of the present work are found in ultracold neutral plasmas, white dwarfs, neutron stars etc.