You can get a 3 carat black diamond on ebay for less than $200. Black diamond is worth much less because it is (99.99% of the time) opaque. Diamonds are sought for their clarity, which the black diamond obviously doesn't have. This is why small black diamonds are used for diamond saw blades; used in construction, and in stone cutting. Until recently (more people appreciating the unique), there has been no marketplace for black diamonds.
Using my knowledge of gemology, I'd say she has got to be shallow, gold digging, and not too informed about diamonds for her to not appreciate the ring.
And any woman who'd break up with a guy because of how much he paid for the ring should end up as an old spinster. My FIL is a wholesaler, my husband got his pick of stones (up to 1 ct) for $1, picking the smaller one because it's clarity was better. Does that mean that he doesn't love me? No, that means that he knows me, and that I'd prefer a smaller, better quality ring as opposed to a larger piece of crap.