The victims were killed over three days by paratroops during Operation Demetrius, when people suspected of paramilitary activity were interned.
The Army said it fired in response to gunfire from republican paramilitaries.
The families said Secretary of State Owen Paterson told them an inquiry was "not in the public interest".
In a statement they said it was "clearly in the public interest" to fully establish the circumstances of the deaths and the role of the Parachute Regiment, "especially given the recent findings of Lord Saville in relation to the events of Bloody Sunday".
"We also refute the suggestion by Mr Paterson that existing processes such as the Historical Enquiries Team will fully answer the families' concerns in relation to these tragic events.
"The Ballymurphy Massacre Families have no confidence in the HET, especially in light of the  
          recent revelations of Dr Patricia Lundy
 in relation to HET's conduct of their review of Royal Military Police investigations into British army killings in the early 1970s."
Last November, Attorney General John Larkin announced new inquests would be held into the Ballymurphy killings. A priest and a mother-of-eight were among those who died.
The families said this was an "important step on our journey for truth" but it would have limitations compared to an independent inquiry.
"It will be able to provide facts and gather crucial forensic, logistical and witness testimony evidence, but it will not be able to examines the causes, context and consequences of the massacre and answer so many of the questions that must be answered," they said.
The families say they want to meet Prime Minister David Cameron, and have called on Taoiseach Enda Kenny to back their campaign as he has done for the family of murdered solicitor Pat Finucane.
Relatives of 10 people shot dead by the army in Ballymurphy in 1971 have said the government has turned down an independent inquiry into the killings.