Iranian protesters set fire to the Saudi embassy in Tehran on Sunday
The Saudi government announced on Sunday that it had broken off diplomatic ties with Iran.
Iran has accused Saudi Arabia of stoking tension in the region.
Saudi Arabia and Iran are the major Sunni and Shia powers in the region respectively and back opposing sides in the conflicts in Syria and Yemen.
The US has appealed for calm, calling for continued diplomatic engagement.
Sheikh Nimr al-Nimr and 46 others were executed on Saturday after being convicted of terror-related offences.
Late on Sunday, police came under heavy gunfire in his home town of Awamiya in Saudi Arabia’s Eastern Province, leaving one civilian dead and a child injured, the Saudi Press Agency said.
Security forces are still hunting the attackers, calling the incident a “terrorist” act, a police spokesman was quoted as saying.
Shia Muslims have complained of marginalisation in Eastern Province.
Saudi Arabia announced it was severing diplomatic relations with Iran after demonstrators stormed the Saudi embassy in Tehran, and late on Sunday gave Iranian diplomats 48 hours to leave.
It has also recalled its diplomats from Tehran.
Iran on Monday accused Saudi Arabia of using the embassy incident to provoke further regional tension, Iranian state TV reported.
It quoted a foreign ministry spokesman as saying: “Saudi Arabia sees its life in pursuit of crises and confrontations and attempts to resolve all of its internal problems by exporting them to the outside.”
Relations between the countries have been strained over various issues in recent decades, including Iran’s nuclear programme and deaths of Iranians at the Hajj pilgrimage in 1987 and again in 2015.
Diplomatic ties were severed between 1988 and 1991.
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 also sparked protests in Iraq, Bahrain and several other countries.