The luxury French fashion brand posted a teaser of the new ad to Twitter on Friday (local time).
The since-deleted tweet promoted the Sauvage campaign as "an authentic journey deep into the Native American soul in a sacred, founding and secular territory".
The extended video, filmed as part of the campaign for the US$120 (NZ$190) fragrance, shows Depp walking through Utah's red rocks, marking his path as a young woman follows him. The actress, Tanaya Beatty, is a Canadian of First Nations descent, the Guardian reports.
The video also captures Native American dancer Canku One Star, a Rosebud Sioux member, performing a war dance on a cliff.
Depp is also filmed strumming a famous guitar riff by Shawnee guitarist Link Wray.