no, because when the wii runs in gamecube mode, it literally becomes a gamecube. And the gamecube came out before the classic controller, therefore does not support it.
The Classic controller was designed so you could play GC and N64 games on the Wii. I have a classic controller and play Super Mario Kart quite happily with it.
The Classic Controller cannot be used to play Gamecube games. The Wii can only provide backward compatibility with the Gamecube controller (notice how the Wiimote turns off while playing Gamecube games), and the Classic Controller lacks the analogue shoulder buttons, so it would be impossible to play any games that made use of them.