There’s certainly no law saying an engagement ring has to be a diamond solitaire, or any kind of solitaire for that matter. But you need to consider durability if you’re thinking of a non-diamond engagement ring. Most women, AFAIK, wear their engagement rings all the time. Choosing a stone that is too likely to be damaged is a bad idea (particularly if you know she’s a klutz who won’t buy an expensive watch for fear of breaking it, like me). There are two issues for gemstone durability- scratch resistance and shatter resistance. Diamonds do well in both of those categories, which is probably one reason why they are popular.
Scratch resistance is measured by Mohs scale of hardness. Here’s a chart showing typical hardnesses of gemstones I’ve heard that sand (which is mostly quartz, with a hardness of 7.0 on that scale) is what’s most likely to scratch a stone in a ring.
Another issue for gemstones is toughness, which is basically resistance to shattering or cracking. It’s independent of hardness- sapphires and rubies are less hard than diamonds (9.0 vs 10.0 on Mohs scale), but are tougher (although diamonds are very good from a toughness perspective as well). There are stones that are high on the hardness scale, but not in toughness- emeralds and topazes are examples. There are also soft stones that are very tough, such as jade.
Apatite is probably not a good choice for an engagement ring. It’s soft (5.0 on Mohs scals), so it’s very likely to get scratched. Its toughness also leaves something to be desired for a ring for wearing every day. An aquamarine would probably be a better choice if she likes the color.
My engagement ring is a sapphire with small diamonds. I also wear a garnet ring on my right hand that Mr. Neville gave me as a wedding present. They seem to have held up pretty well to a few years of the Ultimate Klutz Challenge…