The high court in the northwestern city of Peshawar on May 9 declared the CIA drone strikes targeting suspected Taliban and Al-Qaeda militants to be a "war crime" and ordered Islamabad to take steps to halt them.
Victims' families and their lawyer Mirza Shahzad Akbar have written to Sharif urging him to heed the court's ruling, which calls on the government to take the matter up at the UN Security Council.
"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, seen by AFP.
Akbar said that if Pakistan failed to persuade the US to stop the strikes through the United Nations, "the court has very clearly ordered to shoot down the drones".
Also Read
At a news conference with two relatives of drone victims, Akbar warned that Sharif would face contempt of court proceedings if he did not implement the court order within 14 days.
"My son was 25 years old, he was a labourer and was working in a house with other labourers in the night when the drone strike took place," he told AFP.
"According to tribal law, you kill the son of that person who kills your son, so I will take revenge of my son's killing whenever I have the opportunity."
According to the British Bureau of Investigative Journalism, since 2004 up to 3,587 people have been killed in Pakistan by drone attacks, which Washington says are an effective weapon in the fight against Islamist militancy.