(CNN) The angry demonstrations at the American embassy in Baghdad on Tuesday are just the latest installment in a deepening confrontation between the US and Iran for influence in Iraq.

It's a struggle that dates back to the overthrow of Saddam Hussein in 2003 and the emergence of a Shia-dominated state in Iraq. But in light of the Trump administration's campaign of maximum economic pressure against Iran, it has reached a new and potentially dangerous pitch.

The President himself Tuesday accused Iran of "orchestrating an attack on the U.S. Embassy," and the White House said he would decide "how and when we respond to their escalation."

The protests followed US airstrikes on Friday against a pro-Iranian militia in Iraq -- Kata'b Hezbollah -- that the US holds responsible for rocket attacks against US facilities in Iraq. More than twenty members of the militia were reported killed. The ability of the protestors to breach the outer walls of the embassy compound and set fires suggests a degree of acquiescence on the part of Iraqi security forces.

Trump tweeted: "We expect Iraq to use its forces to protect the Embassy, and so notified!"

Read More