The victims' families had written a letter to the prime minister, asking him to act on the Peshawar High Court's May 9 order that declared the drone strikes a "war crime", lawyer Mirza Shahzad Akbar said.
The court had also directed the federal government to take steps to end the strikes.
Addressing a news conference at the National Press Club, Akbar asked the new government to stop the drones as they were killing people and violating Pakistan's sovereignty.
He claimed 1,400 civilians had died in the drone strikes, which were against the UN Charter and international law.
Several tribal elders and relatives of victims were present at the news conference.
"The court has ordered the government of Pakistan and its security forces to administer a proper warning to the United States that future drone strikes will not be tolerated," Akbar wrote in the letter.
Akbar said if Pakistan failed to persuade the US to stop the strikes through the UN, the High Court "has very clearly ordered to shoot down the drones".
He claimed the prime minister would face contempt of court proceedings if he did not implement the court's order in 14 days.
Pakistan frequently issues statements condemning the drone strikes as a violation of its sovereignty but it has not taken any stronger action to pressure the US to end the campaign that began in 2004.
In his first speech after being elected prime minister by the National Assembly on Wednesday, Sharif called on the US to end drone strikes.
Waliur Rehman, the deputy chief of the banned Tehrik-e-Taliban Pakistan, was killed in a recent drone strike in the lawless tribal belt bordering Afghanistan.
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