The Middle East suddenly hurtled towards a regional conflict over the past week, after Saudi Arabia charged that a missile fired at its capital from Yemen was provided to rebels by Iran and constituted "an act of war".

"Iran cannot lob missiles at Saudi cities and towns, and expect us not to take steps," Saudi Foreign Minister Adel al-Jubeir warned on Nov 6.

"We see this as an act of war."

Saudi leaders also pushed Lebanon's prime minister to resign, as a way of exerting pressure on Hizbollah, the Shi'ite militia and Iran's major ally in Lebanon. "Wherever Iran settles, it sows discord, devastation and destruction," Mr Saad al-Hariri said in his resignation speech on Nov 4, adding that Iran's hands in the region "will be cut off".

These actions underscore a newly aggressive Saudi foreign policy, especially under Crown Prince Mohammed bin Salman, who is eager to challenge Iran more directly.

Can this latest crisis escalate into a military confrontation between Saudi Arabia and Iran? That is possible - mainly because the two regional rivals are already at war, but not on their own territory.

The two powers have been fighting a cold war across the Middle East for over a decade. The proxy battles - in which the two rivals are backing competing factions in Iraq, Syria, Yemen, Lebanon and Bahrain - have shaped the Middle East since the United States invaded Iraq in 2003. While the conflict is partly rooted in the historical Sunni-Shi'ite split within Islam, it is mainly a struggle for political dominance over the Middle East between Shi'ite-led Iran and Sunni-led Saudi Arabia.

Both powers increasingly see their rivalry as a winner-takes-all conflict: If the Shi'ite Hizbollah gains an upper hand in Lebanon, then the Sunnis of Lebanon - and by extension, their Saudi patrons - lose a round to Iran. If a Shi'ite-led government solidifies its control of Iraq, then Iran will have won another round.

These proxy wars, which involve other powers aside from Iran and Saudi Arabia, are at the root of much of the death and destruction in the region over the past six years. They have cost hundreds of thousands of lives, especially in Syria, where more than 400,000 have been killed since the March 2011 uprising against the regime of President Bashar al-Assad.

In March 2015, Saudi Arabia launched a war against Houthi rebels in Yemen, who are supported by Iran. Yemen has now become a central arena of the proxy battle. Saudi officials claimed that the missile which was intercepted on Nov 4 en route to the Saudi capital, Riyadh, had been smuggled into Yemen in parts. Saudi officials said members of Hizbollah and Iran's Revolutionary Guard assembled the missile, and then helped Houthi rebels fire it from Yemeni territory.

In a sign of the expanding proxy war, a Saudi minister said the kingdom also regards the missile attack as an act of war by Lebanon because of Hizbollah's alleged role.

Said Saudi Gulf Affairs Minister Thamer al-Sabhan: "We will treat the government of Lebanon as a government declaring a war because of Hizbollah.

"Lebanon is kidnapped by the militias of Hizbollah, and Iran is behind it."

Since the American invasion of Iraq in 2003, the traditional centres of power in the Arab world - Egypt, Saudi Arabia and other Gulf states - have been nervous about the growing influence of Iran: its nuclear ambitions, its sway over the Iraqi government, its support for the militant groups Hizbollah and Hamas, and its alliance with Syria. The conflict with Iran intensified after the 2011 Arab uprisings.

Saudi leaders became especially nervous when the Arab revolutions spread to Bahrain, a Shi'ite-majority country ruled by a Sunni monarchy only 26km from the Eastern Province, the oil-rich area where a large segment of the population is Shi'ite. The Saudis accused Iran of supporting the Bahrain uprising, and in March 2001 sent more than 1,000 troops to help crush the pro-democracy movement there.

In January 2015, King Abdullah died after 20 years in power, and was succeeded by his brother Salman. Instead of relying on US military intervention and battling Iran through proxies and chequebook diplomacy, as his predecessor had done, the new king and his advisers pursued a more aggressive policy: He launched a war against Houthi rebels in Yemen after only two months in power. King Salman also appointed his then 29-year-old son as defence minister to oversee the campaign.

After two years, Saudi Arabia is bogged down in the Yemen conflict. Despite intensive air strikes and a naval blockade, the Saudis and their allies still have not been able to dislodge the Houthis from Yemen's capital, Sanaa. But as long as Iran and Saudi Arabia see their rivalry as a zero-sum game, there will be no end to the sectarian bloodshed in the Middle East.

• The writer is a journalism professor at New York University and a former Middle East bureau chief at Newsday.