The Minister of State for Skill Development and Entrepreneurship had on Monday said the party was in power to "change the Constitution".
"People agree with the word secular as it is mentioned in the Constitution. It has changed many times before and we have come to change the Constitution," Mr Hegde said at an event in Kuknur in Koppal district, 400 km to the north of the state capital.
Those who call themselves secular are like people "without parentage", the minister had said.
"If someone says I'm a Muslim, Christian, Lingayat, Brahmin or a Hindu, I feel happy because they know their roots. For those calling themselves secularists, I don't know what to call them," the 49-year-old minister said.
Calling Mr Hegde's statements "disparaging", the Congress's state unit said the BJP was "denigrating the ideals" on which the Constitution is based.
"Our country is built on the ideals of secularism and co-existence of people of all castes, creeds, races and religions. Our Constitution is based on ideals, which the BJP is denigrating by allowing Hegde to make such disparaging statements," said Karnataka Congress working president Dinesh Gundu Rao.
Mr Hegde, a resident of Sirsi in Uttara Kannada district, 400 km to the northwest of Bengaluru, was appointed union minister in September after a cabinet reshuffle.
He has been a Lok Sabha lawmaker from Uttara Kannada constituency for five terms.