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.

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."