Plenty of fans take this as Ten’s declaration of love for Rose. He cares about her so much, he couldn’t bear to watch her fade away while he stayed young. It’s a familiar sentiment among immortals: the one scene I remember from the Highlander movie is the opening sequence of the main character’s married life, up to and including his beloved’s final goodbyes. I went there too, on the first pass.
However. “Imagine watching that happen to someone you…” What, Doctor? Someone you love? What do you think every human endures at least once in their life, unless they are fortunate/unfortunate enough to die too young?
Every human watches those they love decay. Every parent dies. Every spouse withers. The Doctor is eternally young; in many ways, particularly this incarnation, eternally a child. He’s seen plenty of death, but somehow aging is more than he can bear. An expression of love it may be, but also an expression of selfishness.
The Doctor leaves his companions behind out of dramatic necessity. Change is a fact of life, in universe and IRL. In many ways, Rose isn’t special at all;* he left Susan and Jo and Sarah Jane, Nyssa and Peri and Grace, and in the end he’ll leave her too. The Doctor moves on.
Still, for this Doctor, it won’t be easy.
* I say this as a fully committed shipper. I believe that Rose IS special to Nine and Ten, though Eleven and Twelve have likely moved on. I think that Ten meant it when he said, “Not you.” This incarnation is just a slice of a very long life, and to him Rose means the world.