A polite way to tell someone to fuck off and die because that person has created the circumstance between you and them which makes you care less if you ever see them again.
He could not deal with her bullshit and destructiveness in their relationship, so he signed the letter "have a nice life."
It basically means goodbye forever. If you say this to another individual, you are telling them, from now on, from this moment on, you couldn't care less what happens to them, as long as you never see, or hear from them again. You are banishing them from your circle of loved ones and friends, from your life. It is almost always justified/warranted.
"Don't worry about responding to this. I just wanted you to know how I felt. Have a nice life."