Why do Virtual Console games appear stretched in widescreen mode?

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Widescreen in 480p as done on Wii, GameCube, PS2, and any other console I know of actually outputs the image in the same 4:3 format as any regular 4:3 image; that's anamorphic widescreen.  The widescreen television then stretches the image horizontally, and it appears to be the correct aspect ratio.  Games that are still 4:3, though, will also still output at 4:3.  If the television isn't told to stop stretching, these titles will appear to be stretched incorrectly.  This is as opposed to widescreen through 720p or 1080i, where the frame is naturally wide, and any 4:3 content will be pillarboxed with blank space (or some other background) on the sides.

Is it possible to compensate for this?  Yes, but at the loss of resolution.  Instead of, say, rendering a game to a full 640x480, it could be rendered to a 480x480 portion of a 640x480 frame and appear to be the correct aspect ratio when horizontally stretched.  But of course this means it's only getting 3/4 as many pixels, so the image loses clarity.

So switch your television back to its 4:3 mode and enjoy the game in the best possible aspect ratio and resolution.

The below image compares Super Mario 64 in a full 4:3 frame with how it would look compensated for a widescreen mode.