Working off the supposition that Hit Points are purely how much damage a creature can sustain before being killed, the Constitution ability can serve as more than just a Hit Point adjustment for hit dice gained when levelling up. Constitution can be used to affect natural healing as well in a definable way.
|Ability Score||Adjustment||Natural Healing||Fast Healing|
Like normal, Adjustment affects the amount of Hit Points per HD the character has. Natural Healing is the amount of hit points a character regains for a day of rest, while Fast Healing is a slower but similar ability of regeneration.
Likewise, Clerical healing magic would work by using the character's Natural Healing rate. The more robust/tough/quick to mend/etc the character is, the more damage healing magic repairs, speeding up the recovery of wounds through holy power, like so:
Cure Light Wounds
Spell Level: Cleric, 1st Level
Immediately cures wounds and injuries equaling 1 point, plus the recipient’s natural healing rate. Optionally, this spell can remove paralysis but when cast in this fashion no damage is cured.
So a Cleric casting Cure Light on a character with an average Constitution (9-12) heals 1+1d3 hit points of damage, while a more robust individual (Constitution 15-16) would be healed 1+1d6 points of damage.