Does a GBA Link Cable Connect Nintendo DS to GameCube?

No. A Game Boy Advance Link Cable cannot connect a Nintendo DS to a GameCube.

Hardware Incompatibility

• Connector: GBA Link Cable uses a 10-pin proprietary plug (GBA/SP/GBA Micro). Nintendo DS uses a different 10-pin arrangement (physically keyed differently) for DS-to-DS comms. They do not mate
• Protocol: GBA uses synchronous serial (SIO) at 256 kbps. DS uses a higher-speed proprietary protocol for multi-card/download play. The GameCube's Serial Port 2 (for GBA Player) expects GBA SIO protocol
• Port: GameCube has Serial Port 2 (for Game Boy Player accessory) and Memory Card slots. No DS-compatible port exists

Actual Connectivity

• GBA ↔ GameCube: Via Game Boy Player (disc + GBA cartridge) or GBA Link Cable (select games: Final Fantasy Crystal Chronicles, Zelda: Four Swords Adventures, Pokémon Colosseum/XD)
• DS ↔ Wii: Wireless (Pokémon Battle Revolution, Pokémon Ranch)
• DS ↔ GameCube: No official hardware or software supports this link.