WASHINGTON—Special counsel Robert Mueller said if he had confidence President Trump didn’t commit a crime, he “would have said so,” suggesting in his first public remarks on the Russia investigation that it was Congress’s job to formally accuse a sitting president of wrongdoing.

Mr. Mueller’s comments on Wednesday squarely put the task of assessing Mr. Trump’s conduct in the lap of a Democratic-controlled House that is divided over whether to open an impeachment inquiry. Mr. Mueller said charging the president with a crime...