I'm not sure, but I think "weakened" can't be healed -- only "sick". I've tried force-feeding weakened villagers, but I'm not sure it helps. The villager feels better after eating, but so do all villagers after they eat, regardless of their strength.

And when you have nothing else to do, you can improve a villager's healing ability by "training" him or her at one of the medicinal plants. Studying those plants (may take several attempts) improves healing just like actual healing another villager does.