Tensions between Iraq and the United States over how to battle the Islamic State broke into the open on Tuesday, as Iraqi officials declared that they would fight on their own timetable with or without American help, and as United States warplanes conspicuously sat out the biggest Iraqi counteroffensive yet amid concerns over Iran’s prominent role.

On Monday, Iraq launched a politically sensitive operation to oust Islamic State militants from Tikrit, the birthplace of Saddam Hussein, without seeking American approval, officials said. Even as Iraq was taking a first step into a bigger battle to oust the Islamic State from the northern city of Mosul, it was also signaling that its alliance with the United States might be more fraught than officials had let on.

American officials, for their part, voiced unease with the prominent role of Iran and its allied Shiite militias in the Tikrit operation. Shiite militia leaders said that their fighters made up more than two-thirds of the pro-government force of 30,000, and that the Iranian spymaster Maj. Gen. Qassim Suleimani was helping to lead from near the front lines. […]