The president said he was reassigning prosecutor-general Abdel Maguid Mahmoud, regarded as a figure from the era of ousted President Hosni Mubarak, as Vatican envoy.
But Mr Mahmoud has said he will not go.
Meanwhile, at least 12 people were hurt in clashes between Mr Mursi’s supporters and opponents in Cairo.
Witnesses said a rally critical of the president was taking place in Tahrir Square on Friday when a crowd of his supporters stormed the stage.
Protesters pelted each other with stones, a hospital official told the official Mena news agency.
The BBC’s Jon Leyne in Cairo says that, 100 days into his term, this is the first time President Mursi has experienced a major demonstration against him.
His opponents are angry at Egypt’s proposed new constitution which they see as too dominated by Islam, he says.
They say the president has failed to live up to his promises.
The counter demonstration in Tahrir Square by the president’s supporters was organised under the pretext of complaining about the recent court verdict, our correspondent adds.
In Alexandria, supporters of the Muslim Brotherhood – the Islamist movement that backs Mr Mursi – gathered ahead of a speech by the president, chanting: “Down with the prosecutor general” and, “the people want the judiciary to be purged”.
The move against Abdel Maguid Mahmoud followed an angry public response to the high-profile acquittal of Mubarak supporters.
Twenty-four people were cleared of organising attacks on protesters during last year’s uprising.
They had been accused of sending men on camels and horses to break up a protest in Cairo in 2011, leaving several people dead.
Many protesters in Tahrir Square blame Mr Mahmoud for the decision to acquit the former officials.
But his colleagues rallied to his support, calling an emergency meeting of a top judges’ club to condemn the president’s decision.
“Egypt’s law on judicial authority offers judges immunity and prevents exiling them from their posts in order to protect their independence from the executive authority,” said senior judge Walid Shafie after the meeting.
The head of the judges’ club, Ahmed al-Zind, was quoted by AP news agency as saying the judiciary would not bow to the “farce”, adding: “The era of tyrants is over.”
The case against the Mubarak supporters is the latest flashpoint between Mr Mursi’s government, elected earlier this year, and figures associated with the Mubarak era.
Those accused included Fathi Sorour and Safwat al-Sherif, former speakers of Egypt’s two houses of parliament.
Prosecutors said Mr Sherif, who was also the secretary general of Mubarak’s National Democratic Party (NPD), had “contacted MPs, members of the NDP and financiers of the party, inciting them to disperse the protests in Tahrir Square by force and violence”.
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