Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal ($alpha$) and tangential ($beta$) restitution. After a formally exact linear stability analysis of the Navier--Stokes--Fourier hydrodynamic equations in terms of the translational ($d_t$) and rotational ($d_r$) degrees of freedom, the transport coefficients derived in the companion paper [A. Megias and A. Santos, Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients, Phys. Rev. E 104, 034901 (2021)] are employed. Known results for hard spheres [V. Garzo, A. Santos, and G. M. Kremer, Phys. Rev. E 97, 052901 (2018)] are recovered by setting $d_t=d_r=3$, while novel results for hard disks ($d_t=2$, $d_r=1$) are obtained. In the latter case, a high-inelasticity peculiar region in the $(alpha,beta)$ parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at $alpha=0.2$ shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity ($alpha=0.7$), however, a good agreement between the theoretical predictions and the simulation results is found.
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