**IMPORTANT**
If you remove divine magic and also whatever healing spells Bards have, your players will be annihilated as soon as they run across a mummy, lycanthrope, vampire, wight, lamia, or anything else that inflicts magical poisons/diseases, potent mundane poisons/diseases, permanent ability damage, ability drain, level drain, negative levels, energy drain, or the like. Also, remember that the Challenge Rating of such creatures is based on the assumption that the party has a cleric of similar level, thus assuming the party can mitigate these nasty effects to at least a small extent.
In fact, without any means of easily preventing or quickly negating any such effects, the world is likely to have already been overrun by such creatures because no one can stop them. Everyone who tries is just that much weaker against the next such creature, and the problem compounds over time. So keep in mind that you need to remove all such creatures from your campaign, or include some non-divine means of negating/preventing such conditions/effects easily and without absurd amounts of spending.
Also, remember that relying entirely on natural healing will not bode well for an adventuring party going through a dungeon environment, where they might not be able to rest for days or weeks at a time to recover naturally. You might allow the Heal skill to restore a small amount of hit points, though, to speed things up just a bit.