"If Nitish Kumar and (Deputy Chief Minister and BJP leader) Sushil Modi do not sack him, we will be forced to expose and reveal his involvement like what we did in the case of Verma," Tejashwi told media.
The RJD leader alleged the minister had close links with jailed main accused Brajesh Thakur.
Congress spokesperson Premchand Mishra said Sharma should resign on his own or Nitish Kumar should force him to quit. "We will intensify our protest for his resignation," said Mishra.
However, ruling JD-U spokesperson Ajai Alok rejected the resignation demand, saying that the CBI is probing the case. "There is no need for the minister's resignation. If he is found guilty in the CBI probe, he would be sent to jail," said Alok.
The BJP is yet to react to the allegations against Sharma.
Manju Verna had resigned on August 8 after media reports of call details revealed that her husband had spoken to the key accused Brajesh Thakur 17 times.
Rashtriya Janata Dal and Congress leaders have alleged that Verma's husband visited the shelter home run by Thakur "regularly."
The Muzaffarpur case came to light when the Bihar Social Welfare Department filed an FIR based on a social audit of the shelter home conducted by the Tata Institute of Social Sciences, Mumbai.
The Patna High Court is monitoring the CBI probe into the shocking rape of 34 minor girls at the Muzaffarpur shelter home, a horror that has caused nationwide outrage.