Guindi's death and footage of police abuse have confronted Islamist President Mohamed Morsi with uncomfortable parallels with the Hosni Mubarak's regime, deposed in an early 2011 uprising fueled by his security forces' brutality.
The United States, a key benefactor of Egypt, called on Morsi's government to investigate charges of police abuse.
"We urge the government of Egypt to thoroughly, credibly and independently investigate all claims of violence and wrongdoing by security officials and demonstrators and to bring perpetrators to justice," said State Department spokeswoman Victoria Nuland.
Guindi's death on Monday after police transferred him unconscious to hospital came days after policemen were filmed beating and dragging a naked man.
The official Al-Ahram newspaper reported on its website that Culture Minister Mohammed Saber Arab resigned in protest over the incident.
But the state news agency MENA quoted a government spokesman who confirmed Arab had tendered his resignation but denied "reports... on the reasons behind his resignation."
In Tanta, witnesses told AFP that police fired tear gas as marchers approached the provincial headquarters following Guindi's burial in the city.
Protesters, joined by residents of the Nile Delta city, then showered police with stones, witnesses said, who responded with further volleys of tear gas and birdshot.
The chief of police the Gharbiya province, where Tanta is located, said the protesters torched an armoured police vehicle and threw petrol bombs at provincial and police headquarters. He said police detained up to nine people.
"The boys who torched the armoured vehicle, we've detained them. About eight or nine," the police chief, Hatem Othman, told the Egyptian ONTV Live channel.
Guindi's lawyers say that after his arrest he was taken to a police camp in Al-Gabal al-Ahmar in Cairo where he was subjected to torture, before being sent several days later to hospital.
The Popular Current, to which Guindi belonged, said the activist died "as a result of torture."
The presidency said in a Facebook statement that it had asked the public prosecutor to probe Guindi's death, stressing there is "no return to rights abuses of citizens and their freedoms... after the January 25 revolution."
The presidency had also said it deplored what it described as the "shocking footage" of police beating and dragging 50-year-old Hamada Saber during clashes outside the presidential palace on Friday.
The interior ministry ordered an investigation into that incident, insisting it upholds human rights.
But images of Guindi's face bruised and battered on a hospital bed and the footage of Ahmed Saber's beating show how little has changed, according to rights groups.
-- Violence and torture 'systematic' --
In a report published on the anniversary of the uprising, the Egyptian Initiative for Personal Rights said police violence was still rampant.
The group said it investigated dozens of cases of torture in recent months, noting "the consistent use of physical and psychological violence against detainees, which in many cases led to death."
Within hours of the news of Guindi's death, furious activists took to social networking sites condemning police brutality and calling for change.
"Youth still tortured & killed in quest for human dignity. Regime oblivious that violence begets violence & brutality is sure to backfire," said leading dissident and former UN nuclear watchdog chief Mohamed ElBaradei.
Tributes have poured in for Guindi on Twitter and Facebook, with activists dubbing him the "new" Khaled Said, a man who was beaten to death by police in 2010 and who became a symbol of the fight against police brutality.