WASHINGTON — When Rex W. Tillerson, the former chief executive of Exxon Mobil, arrived in Washington five months ago to become the secretary of state, his boosters said he brought two valuable assets to a job that had usually gone to someone steeped in government and diplomacy: a long history managing a global company, and deep relationships from the Middle East to Russia that enabled him to close deals.

But his first opportunity to use that experience — as a behind-the-scenes mediator in the dispute between Qatar and Saudi Arabia — has put Mr. Tillerson in exactly the place a secretary of state does not want to be: in public disagreement with the president who appointed him.

Mr. Tillerson tried to position himself as an intermediary and sought for all sides to put their demands on the table. But President Trump openly sided with the Saudis, first on Twitter, then again at a news conference. Mr. Trump called Qatar a “funder of terrorism at a very high level” just as the State Department was questioning whether the Saudis were using the terrorism charge to cover for “long-simmering grievances” between the Arab nations.

Some in the White House say that the discord in the Qatar dispute is part of a broader struggle over who is in charge of Middle East policy — Mr. Tillerson or Jared Kushner, the president’s son-in-law and a senior adviser — and that the secretary of state has a tin ear about the political realities of the Trump administration. Others say it is merely symptomatic of a dysfunctional State Department that, under Mr. Tillerson’s uncertain leadership, does not yet have in place the senior political appointees who make the wheels of diplomacy turn.