The Gulf emirate is his latest stop in a regional tour aimed at reassuring allies after a shock December announcement by President Donald Trump that US troops would be withdrawn from Syria.
Pompeo's remarks follow tensions between the US and Turkey over the fate of Washington's Syrian Kurdish allies in the fight against Islamic State group jihadists.
Turkey had reacted angrily to suggestions that Trump's plan to withdraw troops was conditional on the safety of the US-backed Kurdish fighters, seen by the Turkish government as terrorists. US-led operations against IS in Syria have been spearheaded on the ground by the Kurdish-dominated Syrian Democratic Forces.
Ankara sees the backbone of that alliance, the Kurdish People's Protection Units (YPG), as a terrorist group linked to the Kurdistan Worker's Party (PKK) which has fought a decades-long insurgency against the Turkish state.