(CNN) The fallout of Saudi Arabia's execution of a Shiite cleric is spreading beyond a spat between the Saudis and Iranians, as other Middle East nations chose sides Monday and world powers Russia and China weighed in.

Relations between Saudi Arabia and Iran -- two Middle Eastern powerhouses -- quickly deteriorated following Riyadh's execution of Shiite cleric Nimr al-Nimr Saturday.

Hours after the death sentence was carried out, protesters in Shiite-majority Iran attacked the Saudi Embassy in Tehran. The Saudis cut diplomatic relations with Iran over the attack on its embassy.

Officials from both countries defended their positions Monday and showed no sign of backing down.

Saudi Arabia suspended all flights to and from Iran. It also sent a letter to the U.N. Security Council accusing Iranian authorities of failing in their duties to protect the Saudi embassy.

Jaberi Ansari, a spokesman for Iran's foreign ministry, said his country is committed to protecting diplomatic missions and reiterated that no Saudi diplomats were harmed -- or even present -- during the attack.

He accused Saudi Arabia of "looking for some excuses to pursue its own unwise policies to further tension in the region."

Meanwhile, some nations picked sides while others called for calm.

Here's the latest on where things stand:

Picking sides

Bahrain announced Monday that it was severing diplomatic ties with Iran, citing Tehran's "blatant and dangerous interference" in Bahrain and other Arab countries.

The United Arab Emirates said it was "downgrading" its diplomatic relations with Iran. The UAE recalled its ambassador in Tehran and said it would also reduce the number of diplomats stationed in Iran, according to state news agency WAM. A government statement said the UAE "has taken this exceptional step in light of Iran's ongoing interference in internal (Gulf Cooperation Council) and Arab affairs that has recently reached unprecedented levels."

The diplomatic row spread to Africa, where Sudan -- a majority Sunni Muslim country -- expelled the Iranian ambassador and the entire Iranian diplomatic mission in the country. Sudan also recalled its ambassador from Iran.

The Saudi government announced the Sudanese move, saying Sudan acted because of "the Iranian interference in the region through a sectarian approach."

Russia and China, two of the biggest geopolitical players in the hemisphere, released statements calling for restraint between Iran and Saudi Arabia.

"Moscow is concerned about escalation of the situation in the Middle East with participation of the key regional players," the Russian foreign ministry said Monday. Russia called on the Saudis and Iranians to "show restraint and to avoid any steps that might escalate the situation and raise tensions including interreligious ones."

China's foreign ministry said it is paying close attention to the events and hopes "all parties can remain calm and restrained, use dialogue and negotiations to properly resolve differences, and work together to safeguard the region's peace and stability."

Could the diplomatic war of words boil over?

It had -- even before Saudi Arabia announced its decision to cut ties with Iran, said Fawaz Gerges, chair of contemporary Middle Eastern studies at the London School of Economics.

"Their conflict is playing out on Arab streets big time," he said.

Already the two nations were on opposite sides of conflicts in Syria, Iraq, Yemen, Bahrain and Lebanon. Now, he said, the question is how much worse things might get.

"The situation is extremely volatile between the two most powerful states in the Gulf, Sunni-dominated Saudi Arabia and Shiite-dominated Iran. You have a war of words. You have war by proxies ... This really could get very ugly and dangerous in the next few weeks and next few months," Gerges said.

It's possible a more direct military conflict between Saudi Arabia and Iran could erupt, said retired Lt. Gen. Mark Hertling, a CNN military analyst.

"That's the key issue," he said. "This is spiraling very quickly."

Why are there tensions between Iran and Saudi Arabia?

It's nothing new that the two countries aren't seeing eye to eye.

"Iran and Saudi Arabia are neither natural allies nor natural enemies, but natural rivals who have long competed as major oil producers and self-proclaimed defenders of Shia and Sunni Islam, respectively," University of South Florida Professor Mohsen M. Milani wrote in an analysis for CNN in 2011

Both Saudi Arabia and Iran are painting themselves as victims as tensions between them escalate, Gerges said.

"What you have is not only a clash of narratives, you have basically a huge divide, a war by proxy, a cold war taking place between Saudi Arabia and Iran," he said. "It's a war about geopolitics. It's about power. It's about influence."

So why are things getting worse now?

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It wasn't long before protesters attacked the Saudi embassy in Tehran, hurling Molotov cocktails and cheering as the building caught fire. Iran's Supreme Leader Ayatollah Ali Khamenei called for divine revenge against Saudi Arabia.

"It was almost inevitable that this (the severing of diplomatic relations) would follow, especially since the response from Iran, completely expectedly, was full of rage, and Iran's supreme leader essentially summoned the wrath of God against Saudi Arabia," said Bobby Ghosh, a CNN global affairs analyst and managing editor of Quartz.

But analysts say looking within Iran and Saudi Arabia gives a greater understanding of why both countries have an interest in fueling the rivalry.

"There are domestic reasons for both of these countries right now to refuse to pull punches against each other," said Ian Bremmer, president of the Eurasia Group consulting firm.

Saudi Arabia, he said, is dealing with plummeting oil prices and an internal succession battle over who will next take the throne.

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Iran, he said, needs a way to block reformists and Western advances in light of the recent nuclear deal. For both sides, he said, nationalist behavior can score points at home.

"That," Bremmer said, "makes this an incredibly dangerous conflict."

What can we expect to see in the coming days?

Don't expect the heated rhetoric to die down any time soon, analysts said.

"This is Saudi Arabia saying, 'The gloves are off,'" Ghosh said.

Gerges said that could ripple across the region.

"We were hoping that a diplomatic solution could be found to the Syrian crisis in the next few months. Forget about it," he said.

"We were hoping for a diplomatic solution in Yemen. Forget about it. ... Here, you have the two most powerful Islamic states in the heart of the Middle East now basically waging a direct confrontation, as opposed to an indirect war by proxy, so ... we should be really alarmed at the escalation of the confrontation."

CORRECTION: A previous version of this story incorrectly identified the country that executed Shiite cleric Nimr al-Nimr. Saudi Arabia executed him.