The Iran-backed Huthi rebels also threatened to attack ports and airports in Saudi Arabia and the United Arab Emirates, escalating a crisis between Riyadh and Tehran.
Saudi Arabia's powerful crown prince had accused Iran of supplying missiles to the Huthis, which he said "could be considered as an act of war".
Iranian Foreign Minister Mohammad Javad Zarif retorted that "the allegations by Saudi officials were contrary to reality", a foreign ministry spokesman said.
Washington's ambassador to the United Nations, Nikki Haley, said Tuesday that Iran had supplied a missile to the Huthis that was fired into Saudi Arabia in July, and referred to Riyadh's claim that the weapon used on Saturday "may also be of Iranian origin".
"By providing these types of weapons to the Huthi militias in Yemen, Iran's Islamic Revolutionary Guard Corps is violating two UN resolutions simultaneously," Haley said.
"We encourage the United Nations and international partners to take necessary action to hold the Iranian regime accountable for these violations."
Soaring tensions between the key oil producers pushed crude closer to two-year highs on Tuesday and spooked Gulf markets.
Saturday's attack showed that despite a more than two-year Saudi-led bombing campaign and blockade, the Huthis retain missiles capable of striking targets deep inside the kingdom.
"All airports, ports, border crossings and areas of any importance to Saudi Arabia and the UAE will be a direct target of our weapons, which is a legitimate right," the rebels' political office said in statement.
"We will not stand idly by -- we will seek more radical means to prevent both the tightening of the blockade and all acts aimed at starving and humiliating the people of Yemen."
The missile, intercepted and destroyed near Riyadh international airport, was the first to reach the Saudi capital and underscored the growing fallout for Saudi Arabia and its UAE ally from their involvement in neighbouring Yemen.
The two are the major powers in a coalition that has been fighting the Huthis since 2015 in support of the internationally recognised government.
Since Saturday's attack, the coalition has tightened its blockade of rebel-held areas of Yemen, blocking UN-supervised aid deliveries despite urgent appeals from the world body.
The coalition said it aimed to fill gaps in inspection procedures that enable "smuggling of missiles and military equipment" to the rebels.
But the blocking of aid threatens some seven million people already on the brink of famine.
The UN urged the coalition to lift the blockade as soon as possible.
"If these channels, these lifelines, are not kept open it is catastrophic for people who are already in... the world's worst humanitarian crisis," said Jens Laerke, a spokesman for the UN's humanitarian coordination office (OCHA) in Geneva.
OCHA said the coalition had also asked it to clear ships from the rebel-held Red Sea port of Hodeida, a key entry point for UN aid.
Laerke said fuel prices in rebel-held areas had jumped by up to 60 percent and cooking gas prices had doubled.
The International Committee of the Red Cross also urgently called for humanitarian access to be restored.
"Humanitarian supply lines to Yemen must remain open," said Robert Mardini, who heads ICRC's Near and Middle East operations.
"Food, medicine and other essential supplies are critical for the survival of 27 million Yemenis already weakened by a conflict now in its third year."
That war shows no sign of ending, and Saturday's missile attack has escalated tensions between Riyadh and Tehran, which back opposing sides in wars and power struggles from Yemen to Syria.
On Monday, the Saudi-led coalition in Yemen said it reserved the "right to respond" to the missile attack, calling it blatant military aggression by the Iranian regime.
'Milking their country'
Saudi Arabia is also embroiled in the biggest purge of the kingdom's elite in its modern history.
Dozens of high-profile figures including princes, ministers and billionaire tycoon Al-Waleed bin Talal were swept up in the weekend purge -- just after the creation of an anti-graft commission headed by Prince Mohammed.
US President Donald Trump late on Monday voiced support for the crackdown, saying some of those arrested had been "milking their country for years".
The purge underscores Prince Mohammed's unprecedented reform drive for a post-oil era while consolidating power before his eventual succession as king.
The commission hinted Monday that it could widen its crackdown with further arrest warrants and travel restrictions.