Why didn't Dodge get on the plane?

I think he didn't want to get in the way of her reuniting with her family. He also had no idea if she returned his feelings of true love, the shots of Keira Knightley staring at him longingly were seen by the audience, not the character himself.

It was a Cessna 152, a two-passenger plane. I think they chose that plane specifically so Dodge had to make that choice.