Can You Connect a Nintendo DS to a Wii? No—Incompatible Platforms
No. The Nintendo DS and Wii are incompatible platforms. You cannot plug a DS cartridge into a Wii, nor does the Wii have hardware/software to run DS code.
Why Not?
Different Architecture: DS (ARM9/ARM7) vs. Wii (PowerPC).
Different Media: DS Game Card vs. Wii Optical Disc.
No Video Input: Wii cannot accept a video signal from a DS.
Playing DS Games on a TV
Third-Party Capture/Adapter (Hardware): Devices like the "Katsukity" or "Loopy" capture boards install inside a DS/3DS to output HDMI. Expensive, requires soldering/modding.
Emulation (Software): Play DS roms on a PC/Wii (homebrew) via emulators (DeSmuME, melonDS). Legal only if you own the original cart and dump it yourself.
The intended way to play DS games is on a DS, 3DS, or 2DS family system.