It's more the other way round. Stamina does not increase weapon damage. Weapon damage increases Stamina, or at least restores it, when you make a heavy attack. Most of your slotted skills will be using stamina, and it's those, not the simple weapon attacks, that do the real damage. Some of the skills will do the damage directly, others will add damage to the regular attacks in between. Some will reduce the resistance of the target, which has much the same effect.

With more stamina, you can use the skills more often, and so do more damage, and regaining the stamina with the heavy attacks will help sustain that series. Sets of equipment you use will add stamina, or stamina regeneration, and so make it easier to sustain damage output.

Critical attacks will do more damage, so increasing the chance of a critical attack, with enchantments etc., will increase the total damage done, without affecting the base damage of the attacks.

And don't forget food. Eat something to boost your stamina, and that adds to the pool, too.

Just don't forget to stay alive, too, as you'll do no damage at all if you die.

NB Stealth damage bonus for first strike from stealth -- always a good way to open a fight.

Follow up with a DOT attack to maximize the damage, burn away and execute.

Higher max stam increase damage numbers but not weapon damage.