Disclaimer: I don't own Doctor Who, sadly.
Note: It's written without a specific Doctor in mind, but the site made me choose one, so I went with 11.
He sees her, sometimes.
Well, he says "sees." He doesn't quite see her. After all, he can't do that anymore
(a treacherous part of his mind whispers something about metacrises and beaches and parallel universes)
but he knows she's there, watching, helping. All there is to see is the gold that is so, so familiar, and reminds him of Daleks and game shows and screaming because there's nothing left living for.
("I can see the whole of time and space, every single atom of your existence, and I divide them.")
She saved him. When she became a goddess
(the only deity that Captain Jack ever prays to)
she saw his life and everything in it
("all that ever could be")
and she fixed it – did what she could to give him more of those wonderful days when no one dies.
("Everyone lives, Rose! Just once, everyone lives!")
Sometimes she makes bullets change course, or puts the sonic on the right setting before he reaches for it.
He heard that her song sang Sarah Jane to eternal rest, being there for her when he wasn't. He knows that she does the same thing for Jack, over and over.
(and over and over and over)
In his darkest moments, stuck in a shuttle on a radioactive planet of diamonds, facing Daleks in New York, or in a darkened TARDIS with no one, the shimmering gold flecks are there, in the corner of his eye, making sure he doesn't do something stupid.
(When he hears about a leaf that made an impossible girl possible, he knows that she made that happen, too.)
After all, she loves him, said so herself, and he almost said it back -
("And I suppose, if it's my last chance to say it, Rose Tyler—")
She would probably want him to take care of himself, but since he won't, she did it for him.
("I want you safe. My Doctor.")
Of course, he never really thanked her. Selfish, lonely, old man, he is.
Too bad she couldn't fix that.