How does Fall For You by Second hand serenade relate to Twilight?

The song Fall for you by Secondhand Serenade sounds like something that Edward would of said after she found him in Rome during the second book of the Twilight series "New Moon", and he felt really bad for leaving her. This song shows how Edward will always and forever stay with her. The part of the song that mostly reflects this is as follows...

"This is not what I intended

I always swore to you I'd never fall apart

You always thought that I was stronger

I may have failed, but I have loved you from the start"

This section of the song reprents that Edward was only leaving Bella, because he thought it was what was best for her. It was "not what [he] intended" when Bella was so hurt, that she could hardly move on. He felt that he had failed, but he had always loved her, and he didn't want to hurt her anymore.