https://www.psychologytoday.com/blog/so-sue-me/201508/can-you-really-sue-someone-breaking-you
Yes, it is called breach of promise.
When I was younger I would have said yes. As I've become older my answer would be NO!
Many of the laws on the book are so antiquated,, and should not apply to life these days.
Just for breaking up, as in dating? I say a definitive no. Seems ridiculous to me.