Unarmed Damage affects only damage done with unarmed weapons and fists. This damage is added to the damage value of unarmed weapons, in the case of using fists this is the only damage inflicted on an opponent.
For Melee weapons there is a separate statistic called Melee Damage. Strength does not influence Unarmed damage; it only influences melee damage.
Unarmed Damage is governed directly by the Unarmed skill.