I don't think it was ever questioned if they can or cannot do this. It seems to be more of a question of legitimacy in the public eye. The EP has an interest in not turning itself in a machine for the diffusion of hatred; but the public is invested in knowing what goes on in the chamber too.
But yeah, these things are covered in the EP Rules of Procedure which they themselves draft.
Also, and to be clear, according the the article that first reported on this thing, the powers they already had to delete the record for insulting the institution was invested in the body of presidents (vice-presidents included I imagine).