Arinc said authorities cannot afford to ignore the people and called on "responsible citizens" to stop the demonstrations.
The violence erupted on May 31 after police cracked down on a peaceful rally in Istanbul against plans to cut trees in the city's main Taksim Square.
Protests expanded quickly to Ankara and other cities, with Turks calling for the resignation of Prime Minister Recep Tayyip Erdogan.
At least two people have died during the protests.
Arinc on June 4 apologized to those injured and said the initial protest was based on "legitimate" complaints.
Turkish police and antigovernment protesters clashed again in Istanbul on June 4 as Erdogan remained defiant, rejecting protester demands that he resign and contradicting Arinc by blaming the turmoil on extremists and opposition political forces.
Erdogan on June 3 left Turkey to visit Morocco. In comments there, he predicted that the situation would soon return to calm.
Erdogan said the protests had nothing to do with saving trees but were part of a provocation by his opponents.
"The events [in Turkey] have nothing to do with cutting trees or refurbishing a park. Those who lost at the elections were behind [the demonstrations] and used various ways," he said. "The demonstrations did not take place in every city. The situation is calmer now and people started using their senses in Turkey. Before I go back to Turkey, calm will reign [there]."
Turkey’s NATO ally the United States has again urged authorities to show restraint in dealing with the demonstrations, which have left hundreds of protesters and police injured.
"We are concerned by the reports of excessive use of force by police," U.S. Secretary of State John Kerry said. "We obviously hope that there will be a full investigation of those incidents and full restraint from the police force with respect to those kinds of incidents. And we urge all people involved -- those demonstrating and expressing their freedom of expression and those in the government -- to avoid any provocations of violence."
Turkey’s main stock exchange dropped 10.5 percent on June 3, reflecting investor concerns about the destabilizing effect of the demonstrations.
Protesters accuse Erdogan and his Islamist-rooted movement, which took power in 2002, of becoming increasingly authoritarian.
Some critics accuse of him of trying to force his conservative, religious Islamic outlook on the lives of secular Turks. His government recently, for example, backed legislation curbing the sale of alcohol.
He has also faced criticism over his support for Sunni rebels in neighboring Syria, with some Turks worried the country could become embroiled in the Syrian war.
Erdogan’s party has won three straight elections, curbed the power of the once-dominant military, and encouraged economic growth, enhancing Turkey’s regional influence over the past decade.
Meanwhile, one of Turkey's main trade union confederations is staging a two-day strike in support of the protests.
The left-wing Confederation of Public Workers' Unions (KESK), representing some 240,000 workers, has accused the government of committing "state terror."