I'm not aware that Tolkien ever mentioned this specifically (though there are those on the site who would be better able than I to research that). Keep in mind, however, a couple of things:
- Andúril is, in fact, used in battle. Aragorn uses it against the Wargs and the Orcs in and around Moria, during the Battle of Helm's Deep, in the Battle of the Pelennor Fields, and presumably in the Battle of the Morannon.
- The primary importance of the sword, and the reason for its introduction in the first place, is not as a weapon but as an heirloom. The sword is there because it connects Aragorn with Isildur, the source of his claim to the throne of Gondor; and because it establishes him as a nobleman of Númenorean descent.
It's not unheard of to have duels, or at least one-on-one battles, in epic literature of the type inspiring The Lord of the Rings. Sir Gawain and the Green Knight, which Tolkien translated, has one. Beowulf, which he also began translating, has three. But given that Aragorn uses his sword in battle, and given its primary importance as an heirloom, perhaps Tolkien didn't see a need for a duel scene. He may not even have contemplated it.
As a side note, I'm not sure who would have been an appropriate opponent for him. He couldn't be the one to kill the Witch-king. Tolkien very specifically wanted Éowyn and Merry to have that job. The Mouth of Sauron needed to run away from Aragorn's displayed majesty. I don't know who else would have been available for the job.