Iraqi supporters of pro-Iran factions protested at the embassy on Tuesday, chanting "Death to America," throwing rocks, tearing down security cameras and setting a sentry box ablaze in anger over weekend air strikes that killed two dozen fighters.

It was the first time in years that protesters have been able to reach the US embassy, which is sheltered behind a series of checkpoints in the high-security Green Zone.

"We expect Iraq to use its forces to protect the Embassy, and so notified!" Trump tweeted, saying Iran "will be held fully responsible" for the unrest.

The demonstrators were protesting against US air strikes that killed at least 25 fighters from the hardline Kataeb Hezbollah (Hezbollah Brigades) paramilitary group on Sunday.

The strikes were in response to a 36-rocket attack last week that killed an American contractor at an Iraqi base, the latest in a series targeting US interests in the country.

They have not been claimed, but US security assessments have largely blamed them on Kataeb Hezbollah.

In his tweet, Trump said the US had "strongly responded" to the attack that killed the US contractor, and "always will."

Relations between Washington and Tehran have deteriorated since the US pulled out of the landmark Iran nuclear deal in 2018, imposing sweeping sanctions.

Iraq had long feared being caught in the middle of escalating tensions between the two countries, which are its main allies.

The Iraqi government on Monday denounced the weekend strikes and warned they could affect ties with Washington, which later that day accused Baghdad of having failed to protect US interests.

"It's their responsibility and duty to protect us. And they have not taken the appropriate steps to do so," a senior State Department official said on condition of anonymity.