None of the Oaths really do more damage than the others as such. The only Oath feature that directly impacts your damage is the Oath of Devotion's level 20 ability, Holy Nimbus - it deals 10 radiant damage to any enemies within 60 feet of you. So, to answer your question directly, Oath of Devotion does the most damage.
This is, obviously, a fairly ingenuous answer. The real answer is that you can't look purely for damage.
For example, both the Devotion and Vengeance Oaths have features that increase the accuracy of your attacks, which in the long-term, will increase your damage.
On the other hand, the Oath of the Ancients has an ability that makes you and your allies much tankier and harder to kill. How does that boost damage? Well, you can't deal damage if you're dead, so anything that keeps you alive is also indirectly boosting your damage.
Then there's things like spells. The Oath of Vengeance gives you access to hunter's mark, which potentially lets you add 1d6 damage to every attack you make. But in practice, it's not quite so simple - it requires concentration, and you also have to move it around to make sure it's on the enemy you're attacking. And, obviously, it requires you to make a lot of attacks to maximize the damage it gives.
As a counterexample, the Oath of the Crown gives access to spirit guardians, which deals damage to enemies who come near you - under the right circumstances, this will vastly outdamage hunter's mark. But in the wrong circumstances, it might be completely useless.
So, in the end, there's no easy answer - you'll have to look at all the Oaths and determine which one you think will work best for you.