en.wikipedia.org/wiki/Matt_Stoller

The page is referring to Open Markets' June 27, 2017 statement praising EU antitrust action against Google. New America's archived Open Markets page shows that statement was published on June 27, 2017. On that same date, the European Commission's official press release announced: "Antitrust: Commission fines Google €2.42 billion" in the Google Shopping case.

So the amount in the article is incorrect. A roughly €4.3 billion Google fine was imposed in a different EU case in 2018 involving Android, which appears to be the likely source of the confusion. But the 2017 fine discussed here was €2.42 billion, not €4 billion.

Josh Hawley's own Senate website describes what he did as Missouri attorney general as an antitrust investigation into Google, not a lawsuit. One Hawley press release says he was "the first AG in the country to launch an antitrust investigation into Google," and another says, "I launched a sweeping antitrust investigation of Google when I was Missouri Attorney General."

Missouri's official historical listing shows Hawley served as attorney general from 2017 to 2019 and resigned on January 3, 2019. The major antitrust lawsuit against Google involving Missouri was filed later: the U.S. Department of Justice announced on October 20, 2020 that it, along with eleven states including Missouri, had filed a civil antitrust lawsuit against Google.

So the article incorrectly turns Hawley's investigation into a lawsuit. The more accurate claim would be that Hawley was the first state AG to launch an antitrust investigation into Google.