Dr Ashrawi added: "More than anything else, Israel's impunity must end, and it is time that the perpetrator of the crime (the occupier) is punished and not the victim (the occupied)".
The Palestinians returned to the ICC in the aftermath of accelerated settlement activity in which Israel advanced plans for over 10,000 new homes in 2017.
On the referral of Israel to the ICC, the Palestinian Liberation Organisation said: "The referral will include Israel's past, present and future war crimes and crimes against humanity against the Palestinian people".
Palestinian Authority Foreign Minister Riyad al-Malki arrived on Tuesday at the independent court, based in The Hague, Netherlands, to meet with Prosecutor Fatou Bensouda.
"This referral is Palestine's test to the worldwide mechanism of accountability and respect for global law", he added.
The ICC is a court of last resort - it is authorized to take on cases where national authorities can not or will not launch prosecutions.
The legal maneuver on Tuesday came at a moment of especially high tensions between Israelis and Palestinians after the United States formally relocated its embassy to Jerusalem and the fatal shooting of scores of Palestinians in Gaza whom Israel accused of trying to storm a border fence.More news: Meghan Markle's father successfully undergoes heart surgery, latest report says
The tribunal has been conducting a preliminary probe into crimes in the Palestinian territories since 2015.
It is the eighth time a state party to the court has directly referred a situation to her, she said, warning that a direct referral "does not automatically lead to the opening of an investigation". It is considered a crime against humanity by the ICC and defined by the court as inhumane acts "committed in the context of an institutionalized regime of systematic oppression and domination by one racial group over any other racial group or groups and committed with the intention of maintaining that regime".
"The Palestinians continue to exploit the court for political purposes, rather than work towards resuming the peace process with Israel", a statement said.
While the ICC can indict suspects, it has no police force and has to rely on member states to enforce arrest warrants.
Israel claims east Jerusalem as an inseparable part of its capital - though its annexation is not worldwide recognized.
Under the court's rules, any member can request an investigation.
Under global law it is illegal to transfer populations out of or into occupied territory. Israel also has blamed Hamas for the heavy civilian death toll during the 2014 war.