Doesn't much matter, really...
While I concur with Edlothiad's answer in so far as Gandalf knows his Enemy's heart and Aragorn knows the history of Sauron's treachery, there is one eensy little detail that the answer leaves out:
All of Mouthpiece of Sauron's speechifying is so much of a house of cards built on a foundation of shifting sand being eaten away by the Sea. Sauron does not know where the Ring is, as it wasn't found on Frodo, can not conceive that Gandalf would destroy it nor that Aragorn would actually help. After all, is it not the heirloom of the House of Isildur?
Sauron can only conceive of one possibility: that Aragorn has the Ring and, in his last desperation to seize power from Sauron, has now come to challenge him. Those terms, surely, are meant only to test Aragorn. If he didn't have the Ring after all, Sauron would call his bluff. If he had the Ring, surely he would rebuff Mouthpiece of Sauron.
The battle was, I think, joined simply because Sauron truly feared that Aragorn had the Ring in his possession and would presently storm the Black Tower itself and try to seize Sauron's throne. He'd been humbled before by a Numenorean, recall. That memory must surely sting! But, perhaps if he can throw enough expendable Orcs at Aragorn's "army", then perhaps he could be killed before getting into Mordor proper.
But we know where the Ring actually is! It's on the stoop of Sammath Naur, and three Hobbits of all things are about to contest for its mastery! And we know what happens next: after the brawl, the Ring ends up falling into Fire.
What does this mean?
Ultimately, even if Aragorn had faltered at the last minute and surrendered, surely Sauron would have gloated and would not have missed the opportunity to go up to the Black Gate himself to oversee the surrender & humiliation of Isildur's Heir, the final overthrow of Numenor. With the surrender, Sauron's heart would swell with dark hope: he would know for sure that Aragorn did not have the Ring after all -- why would he surrender with a thing of such great power at his command?
Knowing that his enemies lacked the Ring, he might even feel a sense of relief. He still had time to locate it, after all, it had to be close! He could spend some time, now, figuring out what that Hobbit was doing wandering around the borders of Mordor. So many questions that he did not have time to resolve!
Oh, but wait!
By the time he got up to the Gate, the Ring would have gone over the edge, would have melted, would have died. And with it, Sauron's essence & power. And all the Terms of Surrender uttered by Mouthpiece of Sauron would be moot.