This moment is so beautiful yet so tragic at the same time. He wants to walk out the door, or at least he tries. But he can’t leave without kissing her, since it might be his very last chance to before and if he dies. One last chance to hold her close, to have her lips pressed against his own. As much as he wants to be completely selfless, he can’t. Because he loves her, and he needs to say goodbye. Just in case.
This shows what Elena means for Salvatore brothers. For Stefan, she is independent woman with own choices. He sees her real character and he accept her. For Damon, she is just a girl and the quote “I got the girl.” makes her the thing what he won. It’s really bad use word “got” for girl. She is human, not a toy.