Is There a Halo Game for Nintendo GameCube?

No. The Halo franchise has never been released on a Nintendo console.

Halo is a first-party Microsoft/Xbox Studios IP. The mainline games launched exclusively on:

• Xbox (Halo: CE, Halo 2)
• Xbox 360 (Halo 3, ODST, Reach, 4)
• Xbox One (Halo 5, MCC)
• Xbox Series X|S / PC (Halo Infinite, MCC)

Nintendo platforms (GameCube, Wii, Wii U, Switch) have never received a Halo title. The closest equivalent on GameCube was Metroid Prime (first-person adventure) or TimeSplitters 2 (arena shooter).