Washington asked Tehran to respond "in proportion" after U.S. forces killed top Iranian military commander Qasem Soleimani, the deputy commander of the Revolutionary Guards said.
The death has increased tensions in the region, where the United Kingdom has 400 troops.
What did President Trump say?
United States citizens were meanwhile urged to leave Iraq immediately and American staff were being evacuated from oil fields in the south. American behavior fails to address the many problems in the Middle East and only escalates regional tensions, the two diplomats agreed.
The procession began at the Imam Kadhim shrine in Baghdad, one of the most revered sites in Shiite Islam.
In its statement, the British Foreign Office urged United Kingdom nationals in the region to "remain vigilant and monitor the media carefully".
"The criminal justice process followed in such cases falls below worldwide standards", it said.
Under Soleimani's command, the unit has succeeded in expanding Iran's influence in the so-called Shiite Crescent, a term referring to a crescent-shaped region in the Middle East where Shiite communities reside.
"We will keep this under review".
There are approximately 5,200 U.S. troops deployed across Iraq to help local forces ensure a lasting defeat of jihadists.
Mr Raab has previously urged for a calming of tensions, adding: "Further conflict is in none of our interests".
US-Iranian hostilities have been playing out in Iraq since last week when pro-Iranian militia attacked the US Embassy in Baghdad following a deadly US air raid on the Kataib Hezbollah militia, founded by Muhandis.
Rep. Rick Larsen, D-Everett, said Trump had shared no evidence justifying the attack.
"We have a place on the security council, not so that we can play at being Donald Trump's mini-me or always agreeing with the Americans".
"The Brits, the French, the Germans all need to understand that what we did, what the Americans did, saved lives in Europe as well". We're hurting our allies. "The Americans don't want to invade Iran, if they did that they would have to go onto a war footing".
He added: "The UK can not afford to be neutral if we want to be a serious global player".
Asked if a comment from the Prime Minister could be expected, the Downing Street press office told Yahoo News UK that the government's response was being led by the Foreign Office.
Iranian UN Ambassador Majid Takht Ravanchi wrote that the killing of Soleimani "by any measure, is an obvious example of State terrorism and, as a criminal act, constitutes a gross violation of the fundamental principles of global law, including, in particular, those stipulated in the Charter of the United Nations".
"Following his death, we urge all parties to de-escalate".
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