She said: "It is never an acceptable response to disagreements."
Her call came after a death threat was issued against East Belfast MP Naomi Long, who was warned by police to stay away from her home and constituency office, where loyalists staged demonstrations in protest against a decision by the city council to limit the flying of the Union flag.
She belongs to the non-sectarian Alliance Party, whose offices in Carrickfergus, Co Antrim, and Bangor, Co Down, have also been attacked by loyalists.
Sinn Fein said one of its councillors in Belfast has been threatened in the aftermath of trouble in Belfast, Carrickfergus, and Ballymena, Co Antrim.
Mrs Clinton met Northern Ireland First Minister Peter Robinson and Deputy First Minister Martin McGuinness at Stormont Castle, but the talks on the economy and the peace process were overshadowed by loyalist tensions on the streets and the discovery of another dissident republican bomb in Londonderry, where police arrested four men.
Loyalists took to the streets following the decision to restrict the flying of the flag at Belfast City Hall, where Mrs Clinton and her husband Bill, the then US president, switched on the Christmas tree lights in 1995, a year after the IRA's first ceasefire declaration - and three years before the signing of the Good Friday peace agreement.
With Northern Ireland's two most senior politicians at her side today, Mrs Clinton praised their leadership.
She said peace required sacrifices, compromise and vigilance and the events of the past week showed that the work was not complete.
Mrs Clinton said: "There will always be disagreement in democratic societies, but violence is never an acceptable response to those disagreements. All parties need to confront the remaining challenge of sectarian.
With loyalists threatening another protest in Belfast tomorrow - the busiest shopping day of the year - Mrs Clinton said: "People have strong feelings, but you must not use violence as a means of expressing those strong feelings.
"The only path forward is a peaceful democratic one. There can be no place in the new Northern Ireland for any violence.
"The remnants of the past need to be quickly, unequivocally condemned. Democracy requires dialogue, compromise and constant commitment by everyone to protect the rights of everyone."