This is a sore point with me. The Doctor NEVER apologised to Jack for abandoning him. He explained why he did it. He never said the words "I'm sorry." He only says that when someone is dying.
And yes, the Tardis did try to run away from Jack. It also got the heebie jeebies from Jack's wrongness. But as we have also seen, it got over it. Probably the Doctor did some tinkering to make it accept Jack. The Doctor did offer to take Jack back as a companion, so we can assume that at the latest by that time he had done what was necessary. We have seen Jack with the rest of Dad's army, excuse me, The Doctor's army flying the earth back to it's correct orbit.
The fact remains, and will always remain, that the Doctor abandoned, Jack's word, Jack on a defunct satellite full of corpses above a devastated planet two hundred thousand years in the future with no idea of what had happened to him. Jack, who the Doctor had heard not two hours before say of him, the Doctor, "Never doubted him, never will.", and who then gave his life, with no expectation of resurrection, for the Doctor and all of humanity. What kind of cad would do such a thing?
Please don't anybody give me the "The Doctor was dying and Rose was ill and the Tardis was wonky and he had other things to think about" line. The Tardis is a time machine. The Doctor could have taken off, regenerated, let Rose rest up, (and as we've seen, the went immediately to her house and she was ok) and fine tuned the Tardis to accept Jack, taken all the time he needed, and then popped back in and picked up Jack, two of Jack's minutes after he had left. He could have told Jack, "Sorry, mate, but you are an abomination and the sight of you makes me sick. But I'll take you anywhere you want to go and you can start over." That would have been the honourable course of action. The Doctor didn't take it. He ran away. His words.
I understand that Rose made a reference to going and finding Jack and the Doctor told her that Jack was busy helping rebuild the earth. I did not see that, but if it is so, then the Doctor also lied to Rose.
I confess that there is no Captain save Jack Harkness and that John Barrowman is his Avatar.