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The first time the Doctor asks Rose to dance is in a dimly lit, noisy pub in Croatia. The Torchwood team is celebrating a resolution of tensions with a Corateelian invasion force. It's the Doctor's first field assignment, and the planet is defended peacefully.
They've been back together a month now, slowly but surely feeling their way back to each other (but frustratingly enough, not feeling each other). He thinks she might finally be convinced that he is her Doctor, and he hopes they are closer to making some progress.
"Love Me Tender" comes on the jukebox and the Doctor considers it for all of half a second before he stands up and extends a hand to Rose. "Could I have this dance?" he asks. She smiles, tongue poking out. As they walk to the dance floor hand in hand the team exchanges knowing looks (and a few quid).
Rose rests her head on his chest, feeling that one heart beat and she sighs, contentedly.
"I wish I could've taken you to see Elvis," he murmurs, barely audible over the music and crowd. "I wanted to dance with you to this song."
"You're dancin' with me now," Rose says, brushing a soft kiss to the side of his neck. He feels that kiss all the way down to his toes and back up.
"Yeah...." he says, oh so eloquently. He takes a deep breath and speaks again before nerves get the better of him. "Would you be surprised to know that I hoped that dancing with that night would be followed up...with dancing of another sort?"
She lifts her head off his chest and looks him in the eye. "I would have been very happy with that."
He maintains eye contact, a change from the past when he would have deflected and redirected and the subject would have been dropped. He's ready now to ask a second time.
"I was a bit of a coward then. I don't think I am now. Would you...be ready to dance with me?" he asks.
Rose knows what he means. She pauses long enough for him to begin to backpedal and ramble but she silences him with a kiss. "Yes," she simply says, her lips brushing his. She puts her head back on his chest, and his heart feels like it could burst with joy. He's glad the pub is in their hotel. Once the song ends, they link hands and leave the dance floor, and they immediately head for the lift in the lobby without a word to anyone else. The team had bets on that, too.
Once they get to their room, they dance. It's brilliant and the joy they feel lying in each other's arms afterwards is almost more satisfying than the act itself.
Over time, they find plenty of occasions for both kinds of dancing. One wonderful day, "In the Mood" plays as they spin and twirl around their new TARDIS console.
However, he doesn't ask "Can I have this dance?" again until she's dressed in a white copy of the pink dress she wore to meet Elvis. Instead of pink court shoes she's wearing TARDIS blue Chuck Taylors.
The DJ announces that the bride and groom will have their first dance as a married couple, so the Doctor offers his arm to his wife and asks, "Could I have this dance?"
Rose takes his hand instead of his arm and they walk out to the center of the dance floor. "Moonlight Serenade" begins to play and this time he finds his feet at the end of his legs. They tune out her parents and all the wedding guests. As they glide around the dance floor together, they might as well be the only people in the multiverse.