No. "The Name of the Doctor" indicates that they are are different people.
Also, we know that River (in her Mels incarnation) grew up with her parents Amy and Rory, who were children of the same apparent age as Mels. When the Doctor was shown investigating Clara's history in "The Rings of Akhaten", she had adult-aged parents (Dave & Ellie Oswald).
Clara doesn't resemble any of the known incarnations of River. Since River loses her regenerative ability in "Let's Kill Hitler" and dies in "Forest of the Dead", she cannot regenerate into Clara at any point, either.