Eh, there's many ways he can reveal his name to River (or anyone) and not allow the audience to hear it.
Pretty much how I've thought it would play out. That is whenever whoever/whatever asks "Doctor Who?" they'll have the Doctor say his name, but it will be covered up for the audience over some over the top Murray Gold tune. All the characters present will know the Doctor's name, but the audience won't.
Let's be honest, that's the only way this can turn out.