His two hearts are broken, shattered. The Doctor knows he had no choice; Rose will be better off in the other universe, with her mother. She’ll get to know her father, or the alternate version of Pete Tyler anyway. He seems like a good man.
There’s no way the Doctor can get Rose back anyway, now the breach between universes is closed. As much as it hurts, he wouldn’t if he could, because whether Rose knows it or not, this is for the best. She deserves a normal life and that’s the one thing she can never have with him.