The statement identified Gazal as a member of Hamas' political leadership in the West Bank.
Sami Abu Zuhri, a Hamas spokesperson, said: "The president of the [Palestinian] Authority bears full responsibility for the abduction of Dr Ghazal".
He said Hamas and the Hamas government in Gaza would not succumb to a "policy of blackmail".
"We warn of the grave repercussions of the abduction of Dr Ghazal and the continued detention" of "hundreds of political prisoners" in the West Bank, Abu Zuhri said.
Ghazal was abducted at gunpoint by al-Aqsa Martyres Brigades fighters on Saturday. Later, speaking to Al Jazeera, he said his abductors were demanding that Hamas cease its hunt for those it suspected of triggering an explosion on July 25 that killed five Hamas fighters and a little girl on a Gaza beach..
The kidnapping was first reported by Maan news agency, which said that Ghaza's kidnappers had threatened to execute him unless Hamas ended what they called its anti-Fatah campaign.
One abductor, who identified himself as Abu Qusai, the head of al-Aqsa Martyrs Brigades in Nablus, told Al Jazeera that his group would execute Ghazal within four hours if what he called the Hamas attack on Shijaia neighbourhood and the Hiles family did not stop.
Two Hamas policemen reportedly died when fighting broke out in Gaza City earlier in the day.
Gaza's Hamas-run police said they were attempting to close in on the suspects behind the July 25 bombing.
A medical official said one other person was killed and another 16 people wounded in Saturday's clashes.
In apparent retaliation against Hamas's ongoing security sweep targeting Fatah supporters in Gaza, up to 50 Hamas members, including senior party figures, were detained by the Palestinian security forces in the West Bank over the past week.
Lamis Andoni, Al Jazeera's political analyst, called the latest violence "an extremely dangerous developent" that could trigger an all out Fatah-Hamas confrontation that could have a serious fallout in the West Bank.
"If it is not contained, prospects of a civil will is imminent, especially if the Gaza clashes spark tribal retributions between the Helis clan and clans of Hamas security officers," she said.
The Helis family has announced that it is ready to fight all the way to "defend its families and children".
"You have to decide either be trampled under Hamas's shoes, or stand in dignity," Ahmad Hilles, a Fatah leader and senior clan member, said.
Meanwhile, the Islamic Jihad, a group allied with Hamas, announced that it had begun mediation efforts to stop the fighting and contain the crisis.
Islamic Jihad and other Palestinian factions have in the past succeeded in defusing Fatah-Hamas tensions. But the present crisis could prove far more difficult to resolve, especially if the Helis clan and its supporters declare a war of retribution.
In other news, Hamas officials shut down a radio station affiliated with the Popular Front for the Liberation of Palestine, a statement by the group said.