Disclaimer: I don't own Doctor Who
It was the first time that he had opened himself up to someone after the Time War. The first time that he had allowed himself to hope that he could be accepted by someone after the horrible acts he had committed. Normally, he would just help her and leave; he made a vow to live his life alone after the war as his punishment, but every time he tried to leave her, she continued to chase after him, always searching for answers. Most people would just leave and ignore what was happening, but not her. Not Rose Tyler. "Go home, Rose Tyler. Forget about me." But she didn't. She continued to research him and they met again, and this time her boyfriend was plastic, taken over by the Nestene Consciousness.
When she started crying because her boyfriend might have died, all he really wanted to do was hold and comfort her, but what he did instead was push away so he wouldn't get attached, so she would leave and not get hurt because of him. He put on his tough exterior in the hopes that she would leave. He knew the feeling of having a loved one die, but all he could do was watch as she broke down just like he had done so many times before.
But somehow she still managed to get underneath his skin, the young human from the shop with a spunky attitude who wouldn't take no for an answer. He couldn't stop himself from running with her, from loving every minute of their interactions. However, even though most of him wanted her to travel with him, the rational part that knew of the danger she would be put in kept trying to push her away. He casually mentioned that her boyfriend might have been left alive, carelessly asked her to keep domestics away from him in one last attempt to push her away.
When she saved his life when it would have been so much easier for her to run away and save herself, that was finally when he decided to ask her to travel with him. She was so brave and selfless, she was the exact kind of person he needed to be with. She was one of the few people who might be able to show him how to love again after the massacre he committed the Time War. When she ignored her boyfriend and decided to save him at risk to her own life with her improvisation, that was the very moment he decided to open himself up and take a risk.
He asked her to come with him, to travel all throughout space, anywhere she wanted to go, and she refused, he was devastated that he would still be travelling alone, but he some part of him was happy that she would stay safe at home. She decided to stay and look after her boyfriend, which made him upset, but he knew that she would be out of danger, away from him. As he flew away, though, some part of him just couldn't resist asking one more time, out of desperation to have a friend. "By the way, did I mention it also travels in time?" He stood aside, ready for another rejection, but hoping beyond hope that she would say yes. When she heard her speak, and heard her footsteps running towards the TARDIS, he just managed to contain his glee. Maybe, with the help of this fantastic new human, he would begin to heal from the damage he had caused himself.