WASHINGTON — In the hours after European antitrust regulators levied a record $2.7 billion fine against Google in late June, an influential Washington think tank learned what can happen when a wealthy tech giant is criticized.

The New America Foundation has received more than $21 million from Google; its parent company’s executive chairman, Eric Schmidt; and his family’s foundation since the think tank’s founding in 1999. That money helped to establish New America as an elite voice in policy debates on the American left and helped Google shape those debates.

But not long after one of New America’s scholars posted a statement on the think tank’s website praising the European Union’s penalty against Google, Mr. Schmidt, who had been chairman of New America until 2016, communicated his displeasure with the statement to the group’s president, Anne-Marie Slaughter, according to the scholar.

The statement disappeared from New America’s website, only to be reposted without explanation a few hours later. But word of Mr. Schmidt’s displeasure rippled through New America, which employs more than 200 people, including dozens of researchers, writers and scholars, most of whom work in sleek Washington offices where the main conference room is called the “Eric Schmidt Ideas Lab.” The episode left some people concerned that Google intended to discontinue funding, while others worried whether the think tank could truly be independent if it had to worry about offending its donors.

Those worries seemed to be substantiated a couple of days later, when Ms. Slaughter summoned the scholar who wrote the critical statement, Barry Lynn, to her office. He ran a New America initiative called Open Markets that has led a growing chorus of liberal criticism of the market dominance of telecom and tech giants, including Google, which is now part of a larger corporate entity known as Alphabet, for which Mr. Schmidt serves as executive chairman.

Ms. Slaughter told Mr. Lynn that “the time has come for Open Markets and New America to part ways,” according to an email from Ms. Slaughter to Mr. Lynn. The email suggested that the entire Open Markets team — nearly 10 full-time employees and unpaid fellows — would be exiled from New America.