Why did Abraham stop there?
From the narrative we can see that Abraham was clearly reluctant—out of pure fear, apparently—to question God's judgment.
When asking for 45 in Genesis 18:27, he starts with:
Now that I have been so bold as to speak to the Lord, though I am nothing but dust and ashes
When asking for 30 in Genesis 18:30:
May the Lord not be angry, but let me speak.
And again, asking for 20 in Genesis 18:31:
Now that I have been so bold as to speak to the Lord
And again asking for 10 in Genesis 18:32:
May the Lord not be angry, but let me speak just once more.
He knows he is speaking with the Almighty who is about to show his wrath and while he may not wish the city to be destroyed, he seems afraid to show sympathy for the city as a whole. So instead of mercy for the city—which he might just as easily have asked for—he says let justice be done, but that it wouldn't be just to destroy righteous men along with wicked ones (Genesis 18:23-25).
But Abraham's courage has been holding out. He's asked God to save the city for the sake of fifty, forty-five, forty, thirty, twenty, and ten people—standing up to God six times. Would he have been able to ask for more, perhaps even bring up the idea of redeeming the city, if he had asked a seventh[!] time? After the third plea, Abraham had been asking for ten less each time; the next in the series would be '0'.
Did, perhaps, his courage fail him? Or, according to the explanation I usually hear, did Abraham decide that surely there were ten righteous there, and that he didn't need to ask for more? We can only speculate. But it may not have been up to Abraham at all. It seems that as soon as God answers that he will not destroy the city for ten people, God leaves (Gen 18:32-33):
Then he said, “May the Lord not be angry, but let me speak just once more. What if only ten can be found there?”
He answered, “For the sake of ten, I will not destroy it.” When the LORD had finished speaking with Abraham, he left, and Abraham returned home.
Would God have destroyed Sodom if only one righteous person was found?
God did find one righteous person in Sodom and showed mercy on him, but not the whole city—instead of destroying Lot with the others, he allowed Lot and his family an opportunity to escape the city's destruction.