Is keeping an engagement ring theft in New York City, NY, where he is the one who breaks the engagement, and no fault is claimed against her?
It's not theft, but the donee (recipient) generally is not entitled to keep the ring.
A number of states - not all - have so-called "Heart Balm" Laws, which abolish the ability to sue for a breach of a promise to marry. New York State is one of them.
However, this does not bar actions for recovery of a chattel, money, or securities transferred in contemplation of marriage - including wedding rings.
Consider Mancuso v Russo, 132 AD2d 533 [2nd Dept 1987]. In this case, the plaintiff entered a claim to recover a gift of real property conditional upon subsequent marriage. The Court, and the appeals court, entered judgement in favor of the plaintiff.
The relevant law is found in N.Y. CVR. LAW § 80-b:
NY Code - Section 80-B: Nothing in this article contained shall be construed to bar a right of action for the recovery of a chattel, the return of money or securities, or the value thereof at the time of such transfer, or the rescission of a deed to real property when the sole consideration for the transfer of the chattel, money or securities or real property was a contemplated marriage which has not occurred, and the court may, if in its discretion justice so requires, (1) award the defendant a lien upon the chattel, securities or real property for monies expended in connection therewith or improvements made thereto, (2) deny judgment for the recovery of the chattel or securities or for rescission of the deed and award money damages in lieu thereof
A search for Civil Rights Law § 80-b returns many more cases where claims for (primarily) engagement rings are disputed.