Saudi Foreign Minister Adel al-Jubeir was speaking after demonstrators had stormed the Saudi embassy in Tehran.
Sheikh Nimr al-Nimr and 46 others were executed on Saturday after being convicted of terror-related offences.
Mr Jubeir said that all Iranian diplomats must leave Saudi Arabia within 48 hours.
Saudi Arabia was recalling its diplomats from Tehran, he said.
Earlier, Iran's Supreme Leader Ayatollah Ali Khamenei warned that the Sunni Muslim kingdom would face "divine revenge" for the execution - an act which also angered Shia Muslims elsewhere in the Middle East.
Ayatollah Khamenei called Sheikh Nimr a "martyr" who had acted peacefully.
Media captionAyatollah Ali Khamenei warned events would "catch up" with Saudi Arabia and "give them hell"
Protesters stormed the Saudi embassy in Tehran late on Saturday, setting fire to the building before being driven back by police.
Iran is Saudi Arabia's main regional rival - they back opposing sides in the Syria and Yemen conflicts.
Mr Jubeir said the kingdom would not let Iran undermine its security.
"Iran's history is full of negative interference and hostility in Arab issues, and it is always accompanied by destruction," he told a news conference.
Who was Sheikh Nimr al-Nimr?Image copyrightAFPImage captionNews of Sheikh Nimr's execution prompted an angry response from Shias across the regionIn his 50s when he was executed, he has been a persistent critic of Saudi Arabia's Sunni royal familyArrested several times over the past decade, alleging he was beaten by Saudi secret police during one detentionMet US officials in 2008, Wikileaks revealed, seeking to distance himself from anti-American and pro-Iranian statementsSaid to have a particularly strong following among Saudi Shia youth
Most of the 47 people executed by Saudi Arabia were Sunnis convicted of involvement in al-Qaeda-linked terror attacks over the last decade.
Sheikh Nimr was involved in anti-government protests that erupted in Saudi Arabia in the wake of the Arab Spring, up to his arrest in 2012.
The execution sparked new demonstrations in Saudi Arabia's Eastern Province, where Shia Muslims complain of marginalisation, as well as in Iraq, Bahrain and several other countries.
The top Shia cleric in Iraq, Ayatollah Ali al-Sistani denounced the execution as an "unjust aggression".
The leader of Lebanon's Shia Hezbollah movement, Hassan Nasrallah, launched his sharpest attack yet on the Saudi ruling family on Sunday, accusing them of seeking to ignite a Shia-Sunni civil war across the world.
He said the blood of Sheikh Nimr would "plague the Al Saud [family] until the Day of Resurrection", prompting cries of "Death to the Al Saud!" among an audience watching his address.
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