Just as the title of the post says, can a character gain dex to damage with bows? I am specifically wondering about a fighter. Thanks.
No.
Not in Pathfinder it isn't.
DEX to damage is a very rarely occurring event. Only Slashing Grace, Fencing Grace, and Dervish Dance as feats allow for it. Agile weapons exist, but they are only for melee weapons.
No way to get it on a bow, nor should there be, archery is strong enough as is, ESPECIALLY on a fighter.
Not quite bow, but the Gunslinger archetype Bolt Ace gives Dex to damage with crossbows.
Savage Technologist does it too, for firearms.
Not for a bow. Unchanged rogue gets it for a melee weapon. Trenchfighter and Gunslinger get it for firearms. Bolt Ace gets it for crossbows.
To be fair, fighter archers already do insane damage. Deadly aim, weapon specialization, point blank shot, rapid reload, weapon training, full BAB, strength to damage, plus a bunch of other feats mean you're already doing a lot of damage. You'll want the strength anyway because of your armor's weight.
Not in Pathfinder, no. But in every version of D&D after 3rd/pathfinder, you deal damage with the same stat you use to hit.
You can make a Zen archer that uses a bow with the Guided weapon enchantment. That would allow you to use your Wisdom for both attack and damage.
Not quite the same, but still pretty powerful.
