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The nonlinear propagation of internal gravity waves in the weakly ionized, incompressible Earth's ionosphere is studied using the fluid theory approach. Previous theory in the literature is advanced by the effects of the terrestrial inhomogeneous magnetic field embedded in weakly ionized ionospheric layers ranging in altitude from about $50$ to $500$ km. It is shown that the ionospheric conducting fluids can support the formation of solitary dipolar vortices (or modons). Both analytical and numerical solutions of the latter are obtained and analyzed. It is found that in absence of the Pedersen conductivity, different vortex structures with different space localization can be formed which can move with the supersonic velocity without any energy loss. However, its presence can cause the amplitude of the solitary vortices to decay with time and the vortex structure can completely disappear owing to the energy loss. Such energy loss can be delayed, i.e., the vortex structure can prevail for relatively a longer time if the nonlinear effects associated with either the stream function or the density variation become significantly higher than the dissipation due to the Pedersen conductivity. The main characteristic dynamic parameters are also defined which are in good correlation with the existing experimental data.