The man would have to argue that the engagement ring was given on condition that there was a marriage. That isn't going to get very far! At the time it is handed over it is intended, or hoped, that there will be a marriage but, pace Judge Judy, it is but a token of that hope or intention. There is no doubt that when the man parts with it, he intends it to be hers. As pointed out, we no longer have an action for breach of promise of marriage, which was pursued by jilted women. We can't allow it to be resurrected in favour of an intending bridegroom, by allowing him to claim back a token of his hopes when those hopes are dashed.
Otherwise, he's going to have to argue that there was some contract, for which the ring being bought by him and transferred to her was his consideration ('the price' as it were). There has to be consideration from both sides for there to be a contract; one side paying, or doing, something with the other doing nothing in return is no contract. She has given no 'consideration' in return , so there is no contract. Someone who transfers goods or money to me, voluntarily, for which I am not expected to pay or do anything for, makes me a gift. This ring is a gift.