Initially, the Ninth Doctor was broken and lonely, in such a dark place in his life that I don’t think he ever thought he could come back from it. In the very first episode, he is about to be killed by the Nestene Consciousness and he doesn’t even fight it. He puts up only the smallest of struggles before just accepting his death, when suddenly Rose flies in and saves his life. This small, ordinary girl from the Powell Estates who hadn’t even known for a day jumps in and saves him, and from that moment on you knew that something had changed in him.
When he had first met her, his parting words were “nice to meet you, Rose. Now run for your life.” That’s it. He wasn’t particularily impressed and he didn’t need her to accompany him anywhere. When everything’s over and done with after she saves him, however, he asks her to come with him - not once, but twice. I don’t think he’s ever done that with a companion before (though my Classic Who is very rusty, and if he has done it, it would have been very rarely), asked twice, which should illustrate how much he already needed her.
By the Unquiet Dead, you can tell by the way the Doctor looks at her that he’s already lost, though he isn’t in love with her yet. But there’s just something about Rose Tyler that is reeling him in, that’s attracting him to her and holding him there.
When we get to Aliens of London/World War Three, he’s gone; the fact that he hesitates (“I could save the world but lose you”) is astronomical. The Doctor has always made the touch decisions, always had to sacrifice everything in order to do what is best for the world. Companions have died, people he cared about have been lost, his whole planet burned - all because he has to put the universe before himself and his needs. But in this moment, he doesn’t want to. He wants to be selfish and keep Rose; he can’t lose her. It takes Rose telling him to do it (which that in itself blows him away, that she trusts him so much that she agrees, even though she doesn’t even know what it is) for him to finally make a move. Rose continually surprises him. You have to think about it - the Doctor has been alive for 900 years. I’m sure he’s seen every sort of reaction, every sort of personality, every sort of person, really. But Rose just keeps him guessing.
By Dalek, even an alien who is stripped of its emotions can see that the Doctor is in love with her. In Father’s Day, he allows a paradox to happen simply because he can’t say no to Rose. She means too much to him; all she does is ask and he gives her the universe. I have absolutely no doubt that if anyone else was standing next to him and asked for a do-over, he would have said no, because it was too dangerous. It broke every law of time travel that he knew. But it wasn’t just anyone - it was Rose. And when she caused a massive paradox and almost destroyed the world? Well, she apologized and he forgave her, just like that. He loves her that much and that level of dedication deserves shipping.
Rose’s perspective is different. It takes her much longer to fall in love with the Doctor, partly because she has a boyfriend in the beginning and partly because Nine is harsh in the beginning. He’s condescending and distant, wrapped up in armor. But slowly he starts letting her in and she starts seeing him for who he is. I think in Father’s Day, when she causes the Doctor’s death, she starts to reevaluate him in her life. By The Empty Child/The Doctor Dances, she’s letting her feelings grow (they’re teleported onto Jack’s ship and not even realize it, because they’re so caught up with dancing with each other) and also realizes that the Doctor isn’t asexual, as she previously thought. By Bad Wolf/Parting of the Ways, she’s willing to give up her life to save him. She could have stayed in the past and lived her life in Powell Estates. The Doctor was so far into the future that it would have never affected her. But she didn’t - she fought to get back to him, she ripped apart the TARDIS to save him, she took all of time into her and flew the TARDIS back to him, because she loved him that much.
When Nine transforms into Ten, Ten’s whole world becomes Rose. He was created because of her, created for her; he looks the way he thinks Rose would be attracted to (a pretty boy), he sounds the way Rose does (DT mentioned that there was supposed to be a scene in Christmas Invasion which they explain that his accent is the way it is because of Rose, but it was cut for time), and out of all of the NewWho Doctor’s, Ten is the most human. He essentially has imprinted on Rose. Shannon wrote this wonderful meta about the maturing of the Doctor’s personality:
Series 2 is based around his growing to accept his relationship with Rose, to acknowledge that he can feel humanoid emotions and can want her for himself, and when he loses her in Doomsday the little piece that made him better, that let him escape from the darkness of the Time War was gone. So, yes, he was a complete dick to Martha; but he had at that point lost all hope at finding another woman who symbolized what Rose did – the potential for real happiness. And, because Ten processes everything with such human emotion, even though he was a jackass it’s quite understandable. Finally, Donna offers him a different kind of happiness, which begins to give him hope again, and when Rose returns you can see a glimmer in his face that he may experiencing pure bliss for the first time in ages; but then, when he loses both of them at once, there’s nothing saving him from the darkness of his past, and in the 2009 specials the audience watches him fall to pieces.
Now I’m not saying that the Doctor/Rose is a healthy relationship. By Series 2, it’s clear that Rose and the Doctor need each other too much. They’re entirely dependent on each other and it takes Doomsday to really mature their relationship. They have to grow to live without each other and thus become their own people. The Doctor no longer has the crutch of Rose to lean on and, as Shannon said, ends up withdrawing until he can learn to move on without her. In The Runaway Bride, we know that if Donna hadn’t been there to make him leave, he would have just died, a scene that completely mirrors the Doctor’s actions in Rose with the Nestene Consciousness. Rose, on the other hand, is in a parallel world and is maturing into the confident and strong woman that she always had the potential to be. Furthermore, in Turn Left, she essentially becomes the Doctor and saves the universe. She makes the hard decisions that the Doctor always had to make and is forced to let Donna die, for the good of universe.
When they’re reunited in Journey’s End, their relationships is perfect. They’re both in healthy mindsets and they can be together, because they both know and completely understand each other and themselves. They’re no longer dependent on each other. They know they can live without each other, but instead they will choose not to because they love each other that much and want to spend their lives together.
So that’s why I ship Doctor/Rose. It’s all in the maturing of their characters, of their relationship, that makes them so epic and amazing, that really cements them in my mind as my OTP of all OTPs. Because Rose Tyler and the Doctor love each other, as it should be.