WASHINGTON—Gary Cohn, the White House’s economic policy director, said the Trump administration must do more to condemn hate groups and he did not defend President Donald Trump’s response to violence at a white-nationalist rally in Charlottesville, Va., two weeks ago.

Mr. Cohn said he had come under “enormous pressure both to resign and to remain in my current position” but that he was “reluctant” to leave his post out of a patriotic duty. He addressed the Charlottesville controversy in an interview with the Financial Times...