This would make more sense if she had expressed interest in wearing them before he asked her to marry him, and he'd said no, especially if he'd combined that with telling her that he was saving the rings for a marriage proposal. If that felt like she was taking second place to his ex, I'd understand—not necessarily agree, but the moment someone says "I am keeping my late mother's rings so I can use them if I propose to someone" he's putting symbolism on the history of the rings.

But that "sick to her stomach" reaction indicates something, which could be that she is unusually jealous in general, or that this is her subconscious signalling second thoughts about the engagement for other reasons.

This also ties in with the idea that a woman is supposed to wear, every day of her life, a piece of jewelry selected by someone else, whether or not she likes it. However reasonable her reaction is or isn't, telling her she "should just be okay" with wearing the rings, when he knows she isn't, is overbearing. If he gets to feel however he does (which he does), so does she: and that includes not wanting to wear those rings, but to pick her own.