What was the first Sonic game released on a Nintendo home console?

The first Sonic game released on a Nintendo home console was Sonic Adventure 2 Battle. Originally released for Sega's Dreamcast in 2001, Sonic Adventure 2 was ported to Nintendo's GameCube later that year with the subtitle Battle.

The GameCube version marked a major turning point for Sega. Sega had discontinued the Dreamcast and shifted toward becoming a third-party publisher, making Nintendo a new platform partner rather than a traditional console rival. Battle retained the original game's Hero and Dark story campaigns while adding expanded multiplayer features, extra characters, and Game Boy Advance connectivity.

A common mix-up is Sonic Advance, which also launched in 2001 and was one of the earliest Sonic releases on Nintendo hardware. However, the Game Boy Advance is a handheld, while the question asks specifically about a Nintendo home console. Sonic Adventure 2 Battle therefore holds the canonical answer for the GameCube and for Nintendo home consoles generally.