Nimr al-Nimr, a driving force of the protests that broke out in 2011 in the Sunni-ruled kingdom's east, was among 47 people executed in Saudi Arabia.
Khalaf Abdelsamad, who heads the parliamentary bloc of Iraq's Shiite Dawa party -- to which both Prime Minister Haider al-Abadi and his predecessor Nuri al-Maliki belong -- said Baghdad should take action.
"Abdelsamad urges the Iraqi government to close down the Saudi embassy, expel the ambassador and execute all Saudi terrorists in Iraqi prisons," a statement from his office said.
A justice ministry spokesman said 61 Saudi nationals were detained in Iraqi prisons six months ago. He could not immediately provide a more recent figure.
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"The execution of Sheikh al-Nimr will have serious consequences and bring about the end of the Al-Saud (royal family's) rule," Abdelsamad's office said.
Asaib Ahl al-Haq, one of the most powerful Shiite militias in the country, reacted with similar demands.
"Even as the rulers of Saudi Arabia are supporting terror in the entire world by sending takfiris, weapons and car bombs to Muslim countries, today they executed the most honest man in Saudi Arabia," he said in a statement.
Takfir is a part of the ideology of some extremist Sunni groups -- such as the Islamic State group that controls parts of Iraq and Syria -- that considers other Muslims infidels.
Mohandis has close ties with Tehran, which reacted by warning Saudi Arabia that it would pay a "high price".
Saudi Arabia executed more than 150 people in 2015. According to rights organisations, China, Iran, Pakistan and Iraq are the world's worst executioners.