Al-Aqsa urged Palestinian Authority President Mahmoud Abbas to form a new government "that would not abandon the armed struggle."
The group further urged an end to all contacts with Israel, the Jerusalem Post reported.
Several Fatah officials have called on Abbas to fire Fayad or face a public revolt.
Al-Aqsa said it would renew attacks on Israel in retaliation for the killing of four four Palestinians in Bethlehem this week.
Thousands of Palestinians attended the funeral of the four men in Bethlehem Thursday night.
The deaths created a temporary fusion of political and ideological divisions but particularly enraged Fatah advocates of peace negotiations with Israel.
"The crime committed by Israel against our people aims to blow up the peace process," said Muhammad Khalil al-Laham, a Fatah legislator.
Israel strongly defended the killings as a legitimate response to terrorist acts.