News channels on Thursday showed images of weeping and wailing parents outside the BC Roy Hospital for Children.
A hospital official said the babies were premature, suffering from septicaemia or had low birth weights.
The hospital denied any negligence in remarks to the Press Trust of India news agency.
The investigation into the deaths was announced by West Bengal Chief Minister Mamata Banerjee.
"This is a very sad incident," she said. "We have ordered an inquiry into the unusual deaths."
Around 400 people, including angry parents of the dead children, protested outside the hospital on Thursday and were later reported to have clashed with baton-wielding police.
Correspondents say that the hospital usually accommodates 360 children, but is generally overcrowded, with many infants and young children forced to sleep on the floor because of the lack of beds.
The BBC's Amitabha Bhattasali in Calcutta says that a similar incident at the hospital happened five years ago when 10 children died.