VOTE FIRST BEFORE YOU READ MY RANT ON IT! please. So it would be accurate :)
I love to think that they would do it once. I really like to think it would happen somewhere between episodes 1 and 2 of the second season or right after the satans pit (and considering that it happened shortly before doomsday makes the scene on the beach so much more... Painful. Cuz - the evidence A above).
But the truth is they didn't.
DT said it once during an interview that's for start.
But we can choose not to believe it (cuz creators opinions changes during the years, in 2006 they were like yeah Doctor loved Rose, that was The Love of His Life tm, and then later she became "girl he dared to almost fell in love with")
We know Doctor is asexual. That for one makes it all so much less probable. (Not impossible tho)
About first evidence from above - at first I thought he was like "oh god, u are not having our baby are u?".
But I do not think he could get her pregnant in the first place. And for sure not without them planning this whole process. They are, after all, different spieces. And it is hard to cross breed. He's a genius tho, if she wasn't mortal, and he was willing to start family again (I firmly believe he doesn't want to because of the pain of potentialy loosing everyone again) he would find a way.
People were crossing humans with animals and it was kinda succsessful, and we are mere humans with our tiny technology. Imagine what he could do with that huge brain of his, technology across time and space and his TARDIS.
So yeah now I think that face of his was more of a "you moved on so quickly you have another man and carry his child?". (It is even worse now...)
About evidence B - I personally like to think that Rose was very comfortable around the Doctor. Including being naked or nearly naked in front of him. Not that she would strip on purpose (ya know ~~) but like he walked on on her once, or they went for a swim somewhere, or he had to patch her up or she just went to the console room in her PJ. I think she knew that Doctor was not into sex so she was safe even without clothes. But he was also all about knowing stuff and studing stuff so I can imagine him asking Rose one day to let him study her. They would have a lot of laughs about it.
Also the fact he wasn't into sex doesn't mean he didn't do any physical contact. I bet they hugged a lot. Maybe he even let her sleep on him once or twice. He would read a book or train his patience xD (and isn't that a perfect opportunity to learn someones features?)
Well anyway - she was comfortable and he is a very quick study thats why he knew her so damn well he could make a statue. To me.
I also like to think (and it is my HC) that he wanted to have sex with Rose, but just for her. Like this is for some humans like an ultimate level of intimacy right? Some of them even say a proof of love. And he does love her, undeniably. And we know that he wants all the best for her even if it means without him or will cause him pain or worse. So I can imagine him trying to be very physical with her at some point but failing. Rose would also stop him because she loved him even more and she would never press him into doing something uncomfortable. Not even for her. And Rose definetly would want to go to bed with the Doctor. (Cmon who wouldn't) I bet she had some fantasies about it. I read a fianfic once and the autor wrote something among the lines that "simple act of handholding with him was more exciting than any of her sex life so she was willing to forgo this entirely" just to be close to him. I love this vision and it suits Rose entirely. So I think she wants to, she has fantasies but never acts on them. And he knows about it.
Also... We don't know (or more like I do not, cuz im still fresh in the Fandom :) feel free to inform me on the subcjet) the ways Doctor can be intimate with someone. I said that sex can be viewed as an ultimate form for humans, but he is not one.
We know that he is telepathic, so maybe that is the way for him.
I do not think they fucked in canon. But I do believe (and Im willing to die on that hill) they were very intimate with each other in their own ways.