"The government will not give up the claim of funds which Novartis deprived from the Greek people," Tsipras said.
"We will make use of every power afforded by national and international law to recover the money stolen from the Greek people, down to the last euro," he told his party lawmakers.
Investigators believe that Novartis overcharged the Greek state for medicines in collaboration with government officials who received kickbacks, bilking taxpayers some €3 billion ($3.7 billion) between 2006 and 2015.
Overall, similar practises across the health sector are believed to have cost Greece some €23 billion between 2000 and 2015.
Novartis has issued a statement saying it was "aware of the media reports about our business practices" in Greece and that it was cooperating with the authorities.
Two former Greek prime ministers and eight former ministers have been named by protected witnesses -- whose identity is being withheld -- as allegedly involved in approving overpriced Novartis contracts in return for kickbacks.
Only parliament has the power to investigate former ministers for actions taken during their term in office. The chamber is expected to approve Tsipras' demand for a parliamentary commission to look into the case.
Several of the politicians allegedly named, including former prime minister Antonis Samaras and ex-health minister Dimitris Avramopoulos – now the EU's migration chief – have denied involvement and called on the witnesses to publicly reveal themselves.
Greece's Justice Minister Stavros Kontonis last year said Novartis had likely bribed "thousands" of doctors and civil servants to promote its products.
He also accused Novartis of continuing to sell "overpriced" drugs even after the country was plunged into economic crisis in 2010 and huge cuts were imposed on state budgets, leaving many Greeks without access to affordable medicine.
The case gained attention following a suicide attempt by a Novartis manager in Athens in January 2017.
Novartis was investigated by US authorities in 2014, accused of paying bribes in order to boost sales of some of its medicines, and was later fined $390 million by the US Justice Department.
In March 2017, Novartis also paid $25 million to settle claims involving its Chinese subsidiary.