It came as a relief that River didn't press for more information. If anybody could get him to talk about what he didn't want to talk about, it was River. Maybe later, but not now. The new memories were still too fresh, and stung too deeply, and included his future regenration, or rather his unwillingness to regenerate: he was determined to die.
The Doctor accepted the kiss for what it was - proof that River still loved him, despite all his stupidity and shortcomings... as if that was never in any doubt. But River's warning was taken to his hearts. He nodded slowly to acknowledge that he understood, she was not joking around, but his response was a small, and serious, "I love you, too."