Your teacher is right. The clause does not describe (modify) the story; rather, it tells you what the story is. It is therefore not a relative (or adjective) clause functioning as a modifier, but a content (or noun) clause functioning as a complement.
Relative clauses differ syntactically from content clauses in that the former normally have a gap—a missing element which the relative referent could fill. Take a look at this sentence:
I don't like the story (that) your dad is telling __ to us.
Note that the direct object of the verb TELL is missing but we can safely assume the noun phrase the story tacitly realizes this function.