The war between Mr. Trump and what he calls the “deep state” has entered a new, more volatile phase as the president seeks to assert greater control over a government that he is convinced is not sufficiently loyal to him. With no need to worry about Congress now that he has been acquitted of two articles of impeachment, the president has shown a renewed willingness to act even if it prompts fresh complaints about violating traditional norms.

“The president is entitled to staffers that want to execute his policies, that he has confidence in,” said Robert C. O’Brien, the national security adviser, who supervised Colonel Vindman and his brother, Yevgeny Vindman, also an Army lieutenant colonel, who was dismissed last week from the National Security Council staff even though he did not testify in the House hearings. “We’re not a banana republic where lieutenant colonels get together and decide what the policy is.”

The president’s involvement in Mr. Stone’s case generated vigorous protests and calls for an investigation into whether he improperly sought to skew the prosecution in favor of a longtime associate and adviser. Hours after Mr. Trump’s tweets criticizing the Justice Department for seeking up to nine years in prison for Mr. Stone, the department reversed gears and said it would ask for a lesser sentence.

The Justice Department rejected any link to the president’s tweets, while Mr. Trump insisted that he had nothing to do with the case. But the withdrawal of the four career prosecutors working on the case left the unmistakable impression that they thought something improper had happened.

“The American people must have confidence that justice in this country is dispensed impartially,” Senator Chuck Schumer of New York, the Democratic leader, wrote in a letter asking the department’s inspector general to investigate. “That confidence cannot be sustained if the president or his political appointees are permitted to interfere in prosecution and sentencing recommendations in order to protect their friends and associates.”