The quantum mechanical energy, the Fermi energy, and the degeneracy pressure are derived for a degenerate Fermi gas in D spatial dimensions in the non-relativistic and the extreme relativistic cases. The stability of the system due to the balancing between degeneracy pressure forces and gravitational forces is investigated. We also examine the asymptotic behavior of our results in very large dimensions and their limits in the infinite dimensional space. For computation purposes, we apply these findings to white dwarf and neutron stars.