Is repentance and severing the relationship with a non-Muslim girl, with whom a young man committed a sin short of adultery, obligatory after his family and her family refused their marriage, especially if he feels remorse for having tempted her?

The relationship between a man and a non-mahram (definition) woman is forbidden. You have done well to regret what transpired between you two of forbidden matters, and make that regret a repentance with its conditions. Whoever repents, Allah accepts his repentance, so do not allow room for Satan to make you despair of Allah's mercy. Allah Almighty said: "Say, 'O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.'" And there is no blame upon you for breaking off her engagement, for the principle is the absolution of one's liability, and certainty is not removed by doubt. If it is possible to convince your family to agree to the marriage, then this is a good thing, based on the saying of the Prophet (peace be upon him): "There is nothing like marriage for those who love each other." If marriage is not possible, then look for someone else.