Nike Ching at the State Department and Carla Babb at the Pentagon contributed to this report.
Iranian Foreign Minister Mohammad Javad Zarif suggested Monday that U.S. President Donald Trump "try respect" instead of issuing threats.
He was responding to a Twitter post Sunday in which Trump said: "If Iran wants to fight, that will the official end of Iran. Never threaten the United States again!"
Zarif said Trump, under pressure from a group that includes his National Security Adviser John Bolton and Israeli Prime Minister Benjamin Netanyahu, is hoping to achieve what "other aggressors failed to do.'"
"Iranians have stood tall for millennia while aggressors all gone," Zarif wrote. "Economic terrorism and genocidal taunts won't 'end Iran.'"
Last week, Trump appeared to be backing away from his apparently hawkish stance against Iran, saying he would be open to talks.
When asked by a reporter at the White House on Thursday if the United States was going to war with Iran, Trump replied, "I hope not."
But there has been no apparent let up in the tensions between the United States, its regional allies and Iran.
The State Department says a "low-grade rocket" fell inside the green zone in Baghdad, less than a kilometer from the U.S. embassy Sunday. No injuries or damage were reported.
U.S. Central Command spokesman Capt. Bill Urban said the Pentagon was aware of an explosion outside the embassy, adding, "There were no U.S. or coalition casualties, and Iraqi Security Forces are investigating the incident."
A State Department spokesman says the U.S. will not tolerate such attacks and that it will hold Iran responsible "if any such attacks are conducted by its proxy militia forces."
Saudi Arabia is blaming Iranian-backed Houthi rebels in Yemen for a drone attack on two Saudi oil-pumping stations last week.
The U.S. also suspects Iran was behind the sabotage of four oil tankers off the coast of the United Arab Emirates last week. Two of the damaged tankers were Saudi.
The Saudis also say they will not tolerate Iranian aggression.
"The kingdom of Saudi Arabia does not want war in the region and does not strive for that," foreign affairs minister Adel al-Jubeir said Sunday. "But at the same time, if the other side chooses war, the kingdom will fight this will all force and determination and it will defend itself, its citizens and its interests."
Saudi King Salman has called for emergency summits with Gulf and Arab leaders on May 30 to discuss what the kingdom’s official news agency describes as "aggressions and their consequences."
An Iranian news agency quotes Iran's Revolutionary Guard head Hossein Salami as saying the country does not want war, but is "not afraid" of it.
A statement from the U.S. Navy's Fifth Fleet Sunday spoke of increased maritime patrols and exercises in the Arabian Sea that highlight the "lethality and agility to respond to threat”
The Pentagon has already sent bombers to the region.
The increased tensions with Iran began brewing a year ago when Trump pulled the United States out of the six-nation nuclear deal with Iran.
Under the agreement, Iran limited its uranium enrichment program in exchange for the end of sanctions and economic relief.
The limitations were meant to ensure Iran does not develop nuclear weapons, something Iran denied it had been doing.
Trump, in an interview with Fox News recorded last week and broadcast Sunday, said he does not "want to fight" but that when it comes to Iran, "you can't let them have nuclear weapons."
The reimposed U.S. sanctions have left the Iranian economy in tatters and Iran complains it has yet to see the promised economic benefit from the countries that are still part of the nuclear deal — Britain, China, France, Germany, and Russia.
Iranian President Hassan Rouhani announced two weeks ago he was pulling out of part of the nuclear deal and would restart some uranium enrichment if there were no economic benefits by early July.