While I agree with all who have said, "cocktail party, adults 21 and over" clearly covers this - and your friend/acquaintance has been obstinately obtuse on that point, I diverge on the "you didn't invite him" as the clear demarcation, in this case.
An online invitation which allows guests to reply "plus 1" would, at least among my friends, mean that an unspecified date for a single invitee would be welcome, just include in the response. That may not be true for others, but I think it doesn't matter here if a single guest wanted to bring a child as a date, but the child was outside the the stated guest demographic - he is not 21 or over. I'd stick with that as the reason he will not be welcome as a guest at this event.