This is a somewhat problematic topic. Normally the contrast between "classical" and "romantic" elements in English literature refers specifically to debates between the neo-classical poets of the Augustan period and those of the slightly later Romantic period in the late eighteenth and early nineteenth centuries, long after Shakespeare's death.
Perhaps the best way to discuss this to to understand where the "romance" in the play, between Ferdinand and Miranda, fits within classical notions of genre. This sort of happy romance plot was not a feature of ancient tragedy or Old Comedy (of the sort we see in Aristophanes) but part of Greek "New Comedy" (of which Menander is representative) and the Roman comedies based on New Comedy. The main revenge plot appears to be related to the genre of tragedy, but the fantasitical elements such as Ariel, Caliban, and magic would be more typical of Old Comedy. Thus the romance subplot makes this a mixed genre play, in many ways breaking with classical generic traditions.