Sourani announced his resignation on Saturday to protest against growing security chaos that included bomb attacks against his house and the home of Attorney General Hussein Abu 'Aassi last week. The blasts caused damage but no casualties.
"The president promised to implement all our demands and to afford the appropriate protection and guards for courts and judges," Sourani said after he and Abu 'Aassi met Abbas on Sunday.
Gaza and West Bank judges and lawyers have complained repeatedly of attacks by armed men, who have stormed courtrooms to protest against rulings.
"We hope this time there shall be an end to these attacks," Sourani said.
Sourani said Abbas had reiterated his commitment to an independent judiciary system.
Abbas has pledged to put an end to growing lawlessness in the West Bank and Gaza Strip but has done little to confront armed groups that have become more powerful during a four and a half year Palestinian uprising.