Tarantino is suing the website for copyright infringement.
Gawker Editor-in-Chief John Cook said the company will fight the lawsuit, claiming it merely linked to the 146-page script and did not post the actual document.
However, the suit contends that the link, which was posted on Gawker's Defamer blog, allowed more people to find the script and infringe Tarantino's copyright by downloading and copying it.
Tarantino is also suing an anonymous file-sharing site that is linked in the post for copyright infringement.
Last week, Tarantino blasted the leak in an interview with Deadline.com, saying he was shelving the movie because the script was leaked. It was reportedly a western movie titled "The Hateful Eight."
His lawsuit states he planned to publish the screenplay, and doing so in the past has earned him hefty royalties and advances. His lawsuit states his damages as a result of the Gawker post will be more than $1 million.
The leak was initially limited to a few people, the lawsuit states, and "The Hateful Eight" script did not appear online until after Gawker posted an item encouraging anyone who had a copy to leak it to them.
Tarantino has won two screenwriting Academy Awards for his films "Pulp Fiction" and "Django Unchained." He also received best director Oscar nominations for "Pulp Fiction" and "Inglourious Basterds."
The Associated Press contributed to this report.