The statement underscored the cautious reaction from Iraqi Kurdish leaders who did not condemn neighboring Turkey for an assault on northeastern Syria that has sent thousands of Kurds fleeing. Iraqi Kurdistan relies on Turkish pipelines to export oil and the countries have close political ties.
The Turkish border offensive, following a U.S. troop pullback that in effect gave Turkey a green light, ended Syrian Kurdish rule of "Rojava" - their name for northeastern Syria - and left Iraqi Kurdistan as the Kurds' only self-governed land.
U.S. President Donald Trump's abrupt decision earlier this month to withdraw troops from northern Syria has been criticized in Washington and elsewhere as a betrayal of loyal Kurdish allies who had fought for years alongside U.S. troops against Islamic State.
"We understand the high emotions of our people but the appreciation for the role and contribution of the coalition forces, especially the United States military, should not be affected by undesirable political decisions," Barzani said in a statement.
Outraged that their Syrian kin were betrayed by another U.S. policy decision, protesters in Iraqi Kurdish cities burned Turkish flags last week and some said that America had betrayed the Kurds yet again.
Iraqi Kurds are still reeling from a failed independence bid in 2017. They say the attempt was wrecked by U.S. criticism of their referendum on full Kurdish self-rule, a stance they see as a betrayal by Washington.
The U.S. criticism, plus Turkish and Iranian condemnation, paved the way for Iraqi government forces to retake areas under Kurdish control since Islamic State seized vast parts of Iraq.
U.S. troops had "played a major role" protecting the Kurdistan region over the past three decades, Barzani said.
"They have fought and bled alongside the Peshmerga forces in defending the Kurdistan region."