The primary instability of the wake behind a two-dimensional freely falling circular cylinder is investigated. Numerical simulations are carried out by using the arbitrary Lagrangian-Eulerian (ALE) lattice Boltzmann flux solver, with which the critical Reynolds number Recr for the Hopf bifurcation is identified. Results show that Re_{cr} decreases from 46.1 for a fixed circular cylinder to approximately 42.3 for a light freely falling body with the density ratio of ρ^∗ = 1.01. A series of stability analyses for various degree-of-freedom (DOF) combinations are performed, indicating that the streamwise DOF exerts marginal effects on the primary instability of the wake, whereas the transverse motion can significantly destabilize the wake. In contrast, the rotational motion exhibits a stabilizing effect, thereby increasing the critical Reynolds number Re_{cr}. It is also demonstrated that, as the density ratio increases, the primary instability characteristics gradually converge to those of the fixed circular cylinder, confirming that the larger inertia weakens the structural mobility. Finally, the underlying mechanism is interpreted through the maximum energy of the perturbation velocity k_{max}, which gives a robust scaling law to quantify the primary instability of the wake.