"We will send the injured out for better treatment if needed," Barzani told reporters in Erbil, capital of Kurdistan Region.
He said his party understands the concerns and anger of the people and "we believe in freedom of expression."
Perpetrators changed the aim of the demonstration from peaceful to violent acts that led to an attack on the KDP's headquarters.
"We are not accusing any one now, but the law enforcement forces are investigating to figure out who was behind the attack," he added.
"We are sad about what happened to the headquarters of Goran in Erbil and Soran and we will go to rebuild their offices soon."
He also focused on better relations with other political parties and opposition sides.