...or even start to say 'Sir, about before...' and have Picard stop him and say 'No need, Commander.' Something like that.
Yes, that would be appropriate as well. Probably better than a full on apology.
I watched it again today, and it confirmed my feelings... I *almost* get the feeling that Picard was still offended when he left. The way they shake hands, Picard not really smiling, and then he sort of slowly walks down the stairs without looking back. But he's not the type to look back anyway I guess.