No, different meanings in English. To eat ones words means that someone has been wrong about something. He said he would beat me but I made him eat his words. To break a promise is simply not to do what you have said you would do.

If you have to eat your words, you have to admit you were wrong, often in a humiliating way. Imagine a guy bragging about being the best player on his sports team, but then he gets cut from the team. He would have to eat his words.