(CNN) Iran's government accused Saudi Arabia of "intentionally" striking its embassy in Yemen, another incident likely to ratchet up tensions between Tehran and Riyadh and imperil efforts to forge peace in Syria and combat ISIS.

"Saudi Arabia is responsible for this action, as well as for compensating Iran for injuries to the embassy personnel and the damages to the embassy building," said Iranian Foreign Ministry spokesman Hossein Jaberi Ansari on Thursday, according to state-run IRIB News. "... Iran reserves the right to (legally) pursue this matter."

According to Ansari, the Iranian Embassy in the Yemeni capital, Sanaa, was damaged and several guards were injured in the late Wednesday airstrike.

CNN could not independently confirm whether the embassy was hit and, if so, how severely. The road to the embassy was closed down. Three witnesses in the area around it reported a small explosion Wednesday night but no obvious airstrikes.

And the Saudi-led coalition involved in the Yemeni war pushed back against the claim, with spokesman Col. Ahmed Asseri saying an investigation is needed.

"This is not credible because we have not seen any evidence," Asseri said. "But we will investigate."

Of course, it's doubtful Iran will readily accept the results of a Saudi investigation. Not in light of the longstanding tensions between Sunni-dominated Saudi Arabia and Shiite-led Iran, which have escalated in recent days, threatening an even bigger showdown that could roil not only the Middle East but the world.

Iran bans Saudi imports, pilgrimages to Saudi cities

The Sanaa airstrike, and the war of words that followed, come on the heels of Saturday's mass execution in Saudi Arabia of 47 people , among them prominent Shiite cleric Sheikh Nimr al-Nimr, on terrorism charges.

This act spurred anger in Iran, where protests quickly broke out. The Saudi Embassy in Tehran was attacked and set on fire, prompting Saudi leaders to sever diplomatic ties with Iran.

Photos: Protest in Tehran after Saudis execute cleric Iranian protesters hold portraits of prominent Shiite Muslim cleric Nimr al-Nimr as they confront riot police during a demonstration against his execution by Saudi authorities on Sunday, January 3, outside the Saudi embassy in Tehran, Iran. Saudi Foreign Minister Adel al-Jubeir told reporters that Saudi Arabia is severing ties with Iran after an attack on the embassy. Hide Caption 1 of 10 Photos: Protest in Tehran after Saudis execute cleric Tehran's police chief Gen. Hossein Sajedinia, center right, tries to disperse protesters in front of Saudi Arabia's Embassy in Tehran on January 3. Hide Caption 2 of 10 Photos: Protest in Tehran after Saudis execute cleric Iranian demonstrators burn representations of the U.S. and Israeli flags during a demonstration in front of the Saudi Arabian Embassy in Tehran on January 3. Hide Caption 3 of 10 Photos: Protest in Tehran after Saudis execute cleric Iranian security guards Saudi Arabia's embassy in Tehran while a group of demonstrators gathered to protest on January 3. Hide Caption 4 of 10 Photos: Protest in Tehran after Saudis execute cleric Protesters set fire to the Saudi Embassy in Tehran, Iran, on Saturday, January 2, during a demonstration against the execution of prominent Shiite Muslim cleric Nimr al-Nimr by Saudi authorities. Nimr was a driving force of the protests that broke out in 2011 in Saudi Arabia's east, an oil-rich region where the Shiite minority of an estimated 2 million people complains of marginalization. Hide Caption 5 of 10 Photos: Protest in Tehran after Saudis execute cleric Smoke billows from the windows of the burning Saudi Embassy on January 2. A CNN producer in Tehran said some protesters made it inside the building, setting fire and ransacking some records. Hide Caption 6 of 10 Photos: Protest in Tehran after Saudis execute cleric A firefighter sprays water on the flames on January 2 in Tehran, Iran, a Shiite-majority nation, issued a statement deploring the execution and warning that Saudi Arabia would pay a heavy price for its policies. Hide Caption 7 of 10 Photos: Protest in Tehran after Saudis execute cleric Iranian protesters gather outside the Saudi Embassy in Tehranon January 2. Saudi Arabia said it executed "terrorists" and told Iran to stay out of its internal affairs. Hide Caption 8 of 10 Photos: Protest in Tehran after Saudis execute cleric Several protesters were arrested, police officers at the embassy told CNN. Hide Caption 9 of 10 Photos: Protest in Tehran after Saudis execute cleric There was some damage near the back of the embassy building, witnesses and a police officer said. None of the Saudi diplomatic staff was in the embassy at the time. Hide Caption 10 of 10

Several other nations have sided with Riyadh in this growing conflict, including Bahrain -- where a Sunni monarch rules over a predominantly Shiite nation -- and Kuwait and the United Arab Emirates, both of which recalled their ambassadors to Iran.

Yet Iran has remained defiant, with President Hassan Rouhani on Tuesday lashing out at Saudi Arabia for cutting ties in response to an angry reaction that he called "only natural."

"Of course, the Saudi government, in order to cover up its crime of beheading a religious leader has resorted to a strange measure and has severed its ties with the Islamic Republic," Rouhani said during a meeting with Danish Foreign Minister Kristian Jensen in Tehran

"Undoubtedly, such moves will never hide that great crime," Rouhani added, according to Iran's state-run Press TV.

On Thursday, Iran acted out once more against Saudi Arabia -- this time, economically.

The Iranian Cabinet passed a measure banning the import of all products from Saudi Arabia and prohibiting Iranians from making pilgrimages to the holy cities of Mecca and Medina , the semiofficial Tasneem news agency reported. Millions make pilgrimages each year to those two Saudi cities in rites of passage for Muslims worldwide.

Yemen, like Syria, a proxy battleground for rivals

While Nimr's execution may have been the tipping point, it's far from the lone source of tension between Tehran and Riyadh.

In one respect, this discord dates back 14 centuries to when Sunnis and Shiites split over who should succeed the Prophet Mohammed as Islam's leader.

Today, nearly 90% of the world's Muslims are Sunni. Still, there are powerful Shiite players in the world, including in Iran and Syria, whose embattled leader Bashar al-Assad is Alawite, a Shiite offshoot.

That's one reason Iran has been one of Assad's few notable allies , while Saudi Arabia has given financial aid and weapons to rebels fighting him.

Photos: Unrest in Yemen Photos: Unrest in Yemen The sky over Sanaa, Yemen, is illuminated by anti-aircraft fire during a Saudi-led airstrike on Friday, April 17. The coalition's warplanes have been carrying out strikes against Houthi rebels since President Abdu Rabu Mansour Hadi fled the country in late March. Hide Caption 1 of 42 Photos: Unrest in Yemen A Yemeni boy holds a rifle as Houthi supporters attend a rally in Sanaa, Yemen, on Sunday, April 5, protesting airstrikes carried out by a Saudi-led coalition against Houthi rebels. Hide Caption 2 of 42 Photos: Unrest in Yemen Yemenis dig graves on Saturday, April 4, to bury the victims of a reported airstrike by the Saudi-led coalition in the village of Bani Matar, Yemen. Hide Caption 3 of 42 Photos: Unrest in Yemen Yemenis search for survivors in the rubble of houses destroyed by Saudi-led airstrikes on April 4 in a village near Sanaa. Hide Caption 4 of 42 Photos: Unrest in Yemen Central Sanaa is covered in dust on Friday, April 3. Airstrikes have turned the bustling capital of Yemen into a ghost town. Hide Caption 5 of 42 Photos: Unrest in Yemen A Yemeni man loads a TV set into a van as he prepares to flee Sanaa on Thursday, April 2. Hide Caption 6 of 42 Photos: Unrest in Yemen Militiamen loyal to Hadi take positions on a street in Aden, Yemen, on Thursday, April 2. Houthi rebels seized the presidential palace in Aden, a neutral security official and two Houthi commanders in Aden told CNN. The Houthis are Shiite Muslims who have long felt marginalized in the majority Sunni country. The Sunni Saudis consider the Houthis proxies for the Shiite government of Iran and fear another Shiite-dominated state in the region. Hide Caption 7 of 42 Photos: Unrest in Yemen A framed photo and a purse hang on the wall of a house destroyed by an airstrike near the Sanaa airport on Tuesday, March 31. Hide Caption 8 of 42 Photos: Unrest in Yemen Buildings burn at the Jabal al-Hadid military camp in Aden on Saturday, March 28. Yemeni military officials said an explosion rocked the camp that houses a weapons depot, killing and wounding several people. The camp reportedly had been taken by security forces loyal to former Yemeni President Ali Abdullah Saleh. Some of the forces aligned with the Houthis are also loyal to Saleh, who resigned in 2012 after months of Arab Spring protests. Hide Caption 9 of 42 Photos: Unrest in Yemen Yemeni supporters of the Houthi movement attend a demonstration against Saudi military operations Thursday, March 26, in Sanaa. Hide Caption 10 of 42 Photos: Unrest in Yemen People search for survivors under the rubble of houses destroyed by airstrikes near the Sanaa Airport on March 26. Hide Caption 11 of 42 Photos: Unrest in Yemen Members of Yemen's General People's Committee deploy in Aden, Yemen, on Wednesday, March 25. The militiamen are loyal to Hadi. Hide Caption 12 of 42 Photos: Unrest in Yemen On March 25, honor guards in Sanaa carry the coffins of victims who were killed in suicide bombing attacks several days earlier. Deadly explosions in Sanaa rocked two mosques serving the Zaidi sect of Shiite Islam, which is followed by the Houthi rebels that took over the capital city in January. Hide Caption 13 of 42 Photos: Unrest in Yemen Yemenis stand in front of burning tires during an anti-Houthi protest in Taiz, Yemen, on Tuesday, March 24. Hide Caption 14 of 42 Photos: Unrest in Yemen Medics treat an anti-Houthi protester who was injured during clashes with pro-Houthi police in Taiz on March 24. Hide Caption 15 of 42 Photos: Unrest in Yemen Armed men inspect damage after an explosion at the Al Badr mosque in Sanaa on Friday, March 20. Hide Caption 16 of 42 Photos: Unrest in Yemen A man in Aden holds a police shield that he looted from a base belonging to forces loyal to Saleh on Thursday, March 19. Hide Caption 17 of 42 Photos: Unrest in Yemen Houthi supporters in Sanaa deploy giant national flags Wednesday, March 18, during a demonstration to mark the fourth anniversary of the "Friday of Dignity" attack. In 2011, forces loyal to Saleh opened fire on protesters who had gathered in Sanaa to demand the ouster of Saleh and his regime. Hide Caption 18 of 42 Photos: Unrest in Yemen Supporters of Ahmed Ali Abdullah Saleh, the son of the former President, wave banners and shout slogans during a demonstration in Sanaa on Tuesday, March 10. The demonstrators were demanding presidential elections be held and that the younger Saleh run for office. Hide Caption 19 of 42 Photos: Unrest in Yemen A child raises his fist during a rally by Houthi supporters in Sanaa on Friday, March 6. Hide Caption 20 of 42 Photos: Unrest in Yemen Thousands of armed Yemeni tribal members gather in the southern province of Shabwa on Monday, February 23. Hide Caption 21 of 42 Photos: Unrest in Yemen Supporters of the separatist Southern Movement perform prayers during a demonstration in Aden on Friday, February 13. Hide Caption 22 of 42 Photos: Unrest in Yemen Houthi fighters guard the gate of the presidential palace where a bomb went off and wounded three people in Sanaa on Saturday, February 7. Hide Caption 23 of 42 Photos: Unrest in Yemen Yemeni soldiers guard the presidential palace in Sanaa on Friday, February 6. Hide Caption 24 of 42 Photos: Unrest in Yemen Members of the Houthi movement and their allies attend a meeting in the Yemeni capital on Sunday, February 1. Hide Caption 25 of 42 Photos: Unrest in Yemen Supporters of the separatist Southern Movement flash the victory sign after they seized police security checkpoints on Saturday, January 24, in Ataq, the capital of the Shabwa province in Yemen. Policemen were told to give up their weapons and return to their bases before the militiamen raised flags of the formerly independent South Yemen at the checkpoints. Hide Caption 26 of 42 Photos: Unrest in Yemen Houthi rebels fight with Yemeni protesters during a rally in Sanaa on January 24. Thousands of Yemenis took to the streets of Sanaa in the largest demonstration against Houthis since the Shiite militiamen overran the capital in September. Hide Caption 27 of 42 Photos: Unrest in Yemen On Friday, January 23, Houthis carry coffins of those killed during recent clashes with presidential guard forces in Sanaa. Hide Caption 28 of 42 Photos: Unrest in Yemen A Houthi militiaman sits near a tank near the presidential palace in Sanaa on Thursday, January 22. Hide Caption 29 of 42 Photos: Unrest in Yemen Houthi men wearing army uniforms stand guard on a street leading to the presidential palace in Sanaa on Wednesday, January 21. Hide Caption 30 of 42 Photos: Unrest in Yemen A wounded man rests at a hospital in Sanaa on January 21. He was reportedly injured in fierce clashes the previous day. Hide Caption 31 of 42 Photos: Unrest in Yemen A tank is stationed in front of the Sanaa house of President Abdu Rabu Mansour Hadi on January 21. Hide Caption 32 of 42 Photos: Unrest in Yemen A Houthi rebel mans a checkpoint near the presidential palace on January 21. Hide Caption 33 of 42 Photos: Unrest in Yemen A man walks inside a heavily damaged house near the presidential palace on Tuesday, January 20. Hide Caption 34 of 42 Photos: Unrest in Yemen A woman walks past closed shops in Sanaa on January 20. Hide Caption 35 of 42 Photos: Unrest in Yemen An armed member of the Houthi movement stands guard in the streets of Sanaa on January 20. Hide Caption 36 of 42 Photos: Unrest in Yemen A man surveys his damaged home in Sanaa on January 20. Hide Caption 37 of 42 Photos: Unrest in Yemen Houthis inspect a damaged mosque in Sanaa on January 20. Hide Caption 38 of 42 Photos: Unrest in Yemen Houthi men raise their weapons during clashes near the presidential palace on Monday, January 19. Hide Caption 39 of 42 Photos: Unrest in Yemen Smoke and flames rise in Sanaa during heavy clashes between presidential guards and Houthi rebels on January 19. Hide Caption 40 of 42 Photos: Unrest in Yemen Tribal soldiers protecting the city from Houthi rebels stand guard at the city borders in Marib, Yemen, on January 19. Hide Caption 41 of 42 Photos: Unrest in Yemen Houthi men guard a Sanaa street on January 19. Hide Caption 42 of 42

Staffan de Mistura, the U.N. special envoy on Syria, insisted Tuesday that he's gotten assurances from the Saudis "that the current regional tensions will not have any negative impact" on efforts to resolve the Syrian civil war, which has left hundreds of thousands dead and forced millions to flee. Yet such promises don't necessarily translate to actions -- nor do they speak for Iran's intentions -- especially given all of the other complicating factors that have derailed peace efforts.

Yemen has been another proxy battleground ever since the minority Houthis, who are Shiite, began rebelling last year against the Sunni-led government of President Abdu Rabu Mansour Hadi.

Alongside allies from the UAE, Kuwait, Bahrain, Qatar and elsewhere, Saudi Arabia got involved last March by launching airstrikes in Yemen against rebel targets in support of Hadi.

Those strikes have continued for months, as repeated attempts to forge a truce have failed.

Hadi and his Saudi allies, meanwhile, have accused Iran of giving the Houthis military aid and expertise -- charges that both Iran and the Houthis denied. And on Thursday, the human rights minister in Hadi's government said a U.N. official is no longer welcome in Yemen because of that world body's alleged bias in the Houthis' favor.