Hundreds of men and women in military fatigues and waving flags of Shiite paramilitary forces backed by Tehran marched through checkpoints in the heavily-guarded Green Zone and broke into the embassy's visitor reception building, sparking anger from President Donald Trump and other top US officials.

Protesters carried posters reading: "Parliament should oust US troops, or else we will," and scrawled "No to America!" and "Soleimani is my commander" on the embassy walls, referring to Tehran's pointman for Iraq, Major General Qasem Soleimani of the Revolutionary Guard Corps.

Trump blamed Iran but demanded the Iraqi government protect the embassy compound.

"Iran is orchestrating an attack on the US Embassy in Iraq. They will be held fully responsible," he said via Twitter.

"In addition, we expect Iraq to use its forces to protect the Embassy, and so notified!"

Underscoring the rising crisis in Washington-Baghdad relations, Secretary of State Mike Pompeo pressured Iraqi caretaker prime minister Adel Abdel Mahdi and President Barham Saleh in phone calls to boost embassy security.

Pompeo "made clear the United States will protect and defend its people, who are there to support a sovereign and independent Iraq," the State Department said.

- Easy access -

The assault on the US mission came two days after US air strikes killed at least 25 fighters from the hardline Kataeb Hezbollah (Hezbollah Brigades), a component of the Hashed al-Shaabi paramilitary organization.

The Pentagon said the strikes were in retaliation for the group's killing of an American in Kirkuk.

In response, dozens of Iraqi lawmakers called on the government to review an agreement allowing US troops in Iraq, saying the strikes violated the country's sovereignty.

With Iraq's National Security Advisor Faleh al-Fayyadh -- who is also the head of the Hashed -- among them, the protestors on Tuesday appeared to gain easy access to the Green Zone.

They reached the embassy, throwing rocks, tearing down security cameras and setting a sentry box ablaze in anger over the US strikes.

The Hashed said at least 62 people were wounded after US embassy guards fired an initial volley of bullets, then switched to tear gas and stun grenades to disperse the crowd.

After Abdel Mahdi demanded the protestors leave the compound, some of them pulled back and Iraqi special forces ferried some demonstrators away.

But as a pair of helicopters circled above, the bulk of protestors remained, setting up tents and beginning to cook dinner, signalling a possibly prolonged sit-in.

- US-Iraq ties threatened -

The protest put a focus on the strains in the US-Iraqi relationship as allies of Iran, which enjoys significant support in parts of the Iraqi government, increasingly challenge Washington's influence in the country.

The US has blamed Tehran-backed forces for a series of 11 rocket attacks since October on installations in Iraq where Americans are present.

On Friday, one of those attacks left a US civilian contractor dead, sparking the five US retaliatory airstrikes on Sunday.

"Iran killed an American contractor, wounding many. We strongly responded, and always will," Trump tweeted Tuesday.

Following the deadly strikes, the Iraqi government announced three days of mourning while Kataeb Hezbollah and other pro-Iran factions in the Hashed demanded the "withdrawal of the American enemy".

"American forces acted on their political priorities, not those of the Iraqis," a government statement said.

The attacks "force Iraq to review its relations and its security, political and legal framework to protect its sovereignty", it added.

- Iran oil threatened -

Republican Senator Lindsey Graham, a close confidant of Trump, meanwhile issued a veiled threat to Iran Tuesday that its refining industry could be at risk due to its alleged support for attacks on US assets in the Middle East.

"To the Iranian government: Be careful what you wish for -- a country that depends on the ability to refine oil for its existence needs to be cautious," Graham tweeted.

"President Trump, unlike President Obama, will hold you accountable for threats against Americans and hit you where it hurts the most. Choose your battles wisely," he wrote.