Iran's President Hassan Rouhani has used a speech at the UN to call for an investigation into Thursday's stampede at the annual Hajj pilgrimage to Mecca.
The incident is the deadliest to hit the Hajj in 25 years, with 769 people dead, more than 130 of them from Iran.
Saudi Arabia's Foreign Minister Adel al-Jubeir accused the Iranians of playing politics with a tragedy.
The country's most senior cleric has defended the authorities, saying the stampede was "beyond human control".
King Salman has ordered a safety review into the disaster.
Mr Rouhani described the crush as "heart-rending". As well as the fatalities, 934 people were injured.
But Mr Jubeir, who is also in New York to attend the UN General Assembly, said: "I believe that the Iranians should know better than to play politics with a tragedy that has befallen people who were performing their most sacred religious duty."
Iran's outrage also has political motivations, as its battle with Saudi Arabia for regional influence sharpens week by week, correspondents say.
Analysis: Sebastian Usher, BBC Middle East editor
As criticism of the Saudis intensifies, so has the Saudi response. The intervention of the country's top religious leader is no surprise. Nor is the grand mufti's assertion that fate and destiny are inevitable.
Such fatalism has resonance in the Islamic world, but it won't still a growing clamour of criticism.
Some of those who've been attending Hajj, including survivors of the crush, have been expressing a sense that the Saudis - despite all the billions they've spent on improving the Hajj infrastructure - have not done enough to ensure the safety of the majority of the pilgrims on the ground.
This has been magnified on social media, where the Saudi authorities' placing of the blame on the pilgrims themselves for not following safety instructions has been seen by some as evidence of a lack of sensitivity towards those who come to the Hajj from poorer countries.
Grand Mufti Sheikh Abdul Aziz bin-Abdullah al-Sheikh was visited by Interior Minister Crown Prince Mohammed bin Nayef who is also deputy prime minister and chairman of the Supreme Hajj Committee, on Friday evening, the official Saudi Press Agency (SPA) reported.
"You are not responsible for what happened," the grand mufti said.
"As for the things that humans cannot control, you are not blamed for them. Fate and destiny are inevitable."
The cleric's remarks came after Iran's Supreme National Security Council accused the Saudis of "incompetence" and urged them to "take responsibility" for the deaths.
And on Saturday, Prosecutor General Sayed Ibrahim Raisi said that Iran would seek the trial of the Saudi royal family over its "crimes" in "international courts", Isna news agency reported.
Saturday was the final day of the Hajj, with no further serious incidents reported.
The crush occurred on Thursday morning as two million pilgrims were taking part in the Hajj's last major rite.
The pilgrims throw seven stones at pillars called Jamarat, which stand at the place where Satan is believed to have tempted the Prophet Abraham.
With temperatures around 46C, two massive lines of pilgrims converged on each other at right angles at an intersection close to the five-storey Jamarat Bridge in Mina, a large valley about 5km (3 miles) from Mecca.
It is also the second disaster to strike in two weeks, after a crane collapsed at the Grand Mosque in Mecca, killing 109 people.
Deaths reported so far by nationality
- Iran: at least 134
- Morocco: 87 (media reports)
- Cameroon: at least 20
- Niger: at least 19
- India: 18
- Egypt: 37
- Pakistan: 11
- Chad: 11
- Somalia: 8 (media reports)
- Senegal: 5
- Algeria: 4
- Tanzania: 4
- Turkey: 4
- Indonesia: 3
- Kenya: 3
- Nigeria: 3
- Netherlands: 1
- Burundi: 1
- Other nationalities (numbers not yet known): Benin
Saudi helplines: 00966 125458000 and 00966 125496000