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No one asks him about how he feels about River. He doesn't offer his thoughts on the matter often, so most people who know the truth forget his opinion, except River herself. She asks her father how he feels about the whole situation one evening, after she pops up and Amy's not home. He knows this is his daughter, and the way she asks makes him want to tell her, so he does.
She's family. At the end of the day, even though he knows River is Mels and Melody and River all rolled into one, she's family. Is he angry he didn't get to raise her? Of course. Is he still accepting that his daughter was one of his best friends growing up? Yes. Would he rather have a normal family? Sometimes. But he's proud of his daughter just the same, and he loves her very much, and he knows deep down that while his family may be strange, it is far far better than not having one at all.