Sargsyan had long promised not to run for either president or prime minister after the end of his 2018 term, but changed his tune in March this year.
Prosecutors have opened several criminal cases in connection with the protests, the prosecutor general's office said today.
At least 29 were arrested and several were injured in clashes between protesters and security forces.
Opposition politicians say the shift to a parliamentary republic with a powerful prime minister has been created to increase Sargsyan's grip on power in the impoverished Moscow-allied country.
Serzh Sargsyan's appointment on Tuesday has been accompanied by a wave of mass protests that erupted on Friday.
Protests in Armenia have continued for the fifth straight say shutting Yerevan's downtown.
Ambedkar's birth anniversary celebration today
While Rahul Gandhi tweeted paeans - "Paying rich tribute to Baba Saheb Ambedkar on his birth anniversary". He ensured protection of your rights.It is the duty of the government to take care of your rights.
"I call on all of the population of Yerevan and other towns and cities in Armenia to take to the streets tomorrow (Tuesday), block the main axes, and start a strike", Nikol Pashinian told a rally.
After Sargsyan was first elected president in February 2008, 10 people died in bloody clashes between police and supporters of the defeated opposition candidate.
On one central square, a deputy police chief handed protest leader Nikol Pashinian a notice warning that demonstrators had violated legislation on public gatherings and that police had made a decision to stop the demonstrations. Police attempted to prevent demonstrators from reaching the parliament building, local media reported, while the health ministry said 46 people including six police officers required medical assistance after the clashes. Human Rights Watch called on the Armenian government to respect the right to peaceful assembly.
"The time has come to liberate Armenia's citizens".
Regional analyst Karena Avedissian told OC Media that in these protests, unlike in previous ones, anger has been more directed towards Serzh Sargsyan. "If I am elected, it won't be the third government term of Serzh Sargsyan, but rather the first term of a Republican Party of Armenia government in a parliamentary Armenia", he said. The Prime Minister became the key official in the state, which evoked the outrage of political activists.