All Wii games must support 4x3. Additionally, some percentage of Wii games support outputting an anamorphic widescreen (16x9) signal.
If one tells the Wii that one is using a 4x3 display, and one puts one's 16x9 display in 4x3 mode (with black or grey bars on the side), everything will appear correct at all times. If one tells the Wii that one is using a 16x9 display, and one puts one's 16x9 display in 16x9 mode, everything will appear correct for games which support 16x9 mode. For games which only support 4x3 mode, everything will be stretched horizontally, appearing "too fat".
There is no automatic aspect ratio selection or adjustment, so Wii owners using the Wii in 16x9 mode must manually tell the TV to use 4x3 or 16x9 mode depending on the capabilities of the game being played if they object to aspect ratio distortion.