You are correct; "Whose your daddy" is not grammatical.
Perhaps the confusion is due to the pronunciation similarity between the two phrases, which may have caused your friend to associate it with the incorrect transcription. This phenomenon is also seen when distinguishing "your" and "you're", as noted in "Your" vs. "you're": Why the confusion? .
Edit to quote sources and provide explanation: "Whose your daddy" doesn't make sense because "whose" is a possessive adjective, or a pronoun, as in
- Whose book is this? (adjective)
- Whose is this? (as pronoun)
Any complete, grammatical sentence with these forms requires a verb; in those two examples, the verb is "to be". In Whose your daddy?, there is no verb, and hence it is not a sentence. The second form, Who's your daddy?, contains a verb "is", contracted with a pronoun "who", to form "who is your daddy?" or "who's your daddy?"
Another useful way to look at this is to imagine replacing "your daddy" with the pronoun "this"; such a noun-to-pronoun replacement is always legal. The first "sentence" would then become "whose this?", which doesn't make sense. The second would become "who's this?", which is a perfectly valid question, often asked to a person on the opposite end of a phone call.