No, Metallica is not a goth band, because they play Thrash Metal, while goth music consists of Gothic Rock and its derived genres, like Deathrock, Darkwave, Ethereal Wave, and Coldwave.

If you want to dive into the goth music scene, some of the bands that might appeal to you if you enjoy Metallica are Dronning Maud Land, Fields Of The Nephilim, and Opened Paradise.