Heavy-ion beams, each with current in the kiloampere range and particle energy in the giga-electronvolt range, must be focused onto a millimetre-size spot to provide the power required for ignition of high-gain targets for inertial confinement fusion. However, the focal spot size is always enlarged by chromatic aberration generated by the thermal spread of the beam ions in the direction of beam propagation. Enlarged focal spot degrades the target performance. For high-current beams, the conventional remedy for chromatic aberration using sextupole magnets has been shown to be ineffective. If novel correction schemes can be found, then the spot size can be reduced to below that previously believed possible. Smaller spots can mean lower energy targets so that the heavy-ion fusion (HIF) scenario can look more attractive. Success in laser cooling of ion beams in storage rings has inspired us to explore the feasibility of applying laser cooling for HIF, and the recirculator configuration proposed for HIF appears to be well suited for this purpose. However, using particle-in-cell simulations and theoretical arguments, we demonstrate in this paper that although laser cooling of heavy-ion beams is feasible in principle, the rapid velocity-space diffusion of ions in the bump-in-tail distribution, set up by the cooling lasers, limits the velocity-space compressibility of the thermal spread along the beam. Consequently, laser cooling is impractical for high-current, heavy-ion beams for the proposed recirculator configuration. Nevertheless, if the recirculator architecture or the target requirement can reduce the beam current, then the cooling scheme described here would be useful. This scheme may also be applicable to the RF linac and storage ring approach to HIF.