  Within the context of Rastall gravity, we investigate the hydrostatic
equilibrium and dynamical stability against radial pulsations of compact stars,
where a free parameter $\beta$ measures the deviations from General Relativity
(GR). We derive both the modified Tolman-Oppenheimer-Volkoff (TOV) equations
and the Sturm-Liouville differential equation governing the adiabatic radial
oscillations. Such equations are solved numerically in order to obtain the
compact-star properties for two realistic equations of state (EoSs). For
hadronic matter, the fundamental mode frequency $\omega_0$ becomes unstable
almost at the critical central energy density corresponding to the maximum
gravitational mass. However, for quark matter, where larger values of
$\vert\beta\vert$ are required to observe appreciable changes in the
mass-radius diagram, there exist stable stars after the maximum-mass
configuration for negative values of $\beta$. Using an independent analysis,
our results reveal that the emergence of a cusp can be used as a criterion to
indicate the onset of instability when the binding energy is plotted as a
function of the proper mass. Specifically, we find that the central-density
value where the binding energy is a minimum corresponds precisely to
$\omega_0^2 =0$.
