I'll throw my weight as an English teacher behind Wendy's statement. Anna, 'he smiled' is not a speaker tag, so it cannot be attached to a quotation with a comma. In English, we only allow speaking verbs (and there are certainly over 100 of them--I used to collect them when I was a teenager) to be used as speaker tags. To use your example:
He smiled. "Nice to meet you." means that the speaker says the words with a smile on his lips. There is no need to try attach 'he smiled' to the end of the quotation as a mock speaker tag to get the same effect.
If you're going to use a tag that the reader isn't going to notice - why not just not use anything at all? It would have the same result.
LabRat, the main reason for using "said" is to give the writer a reason to insert the character's name next to his/her dialog, to remind the reader who is speaking. Naked dialog wouldn't give the same effect because the reader might still be counting responses since the last beat or speaker tag to determine who is speaking. Victoria Holt was terrible about that. She would have two pages of dialog without a single speaker tag or beat, and invariably, I'd get lost at some point and have to resort to counting speeches to figure out who said what. Since the word "said" isn't supposed to show up much, the only thing the reader notices is the character's name or pronoun. At least, that's the reasoning.
