Singh said, "My intention was not to hurt anyone, if my comments have hurt anyone, I apologise."
Giriraj Singh had kicked up a row with his remarks asking whether Congress would have accepted Sonia Gandhi's leadership had she not been white-skinned.
As soon as the house assembled for the second part of the budget session after a month-long break, Congress MP Jyotiraditya Scindia raised the matter.
"People are making statements against women. Central minister Giriraj Singh had made a derogatory comment about Sonia Gandhi. This kind of comments are not only against the women of the country, but the entire country should hang their heads in shame.
"We not only want an apology, but also resignation of the minister. The prime minister should also apologize," he said.
Parliamentary Affairs Minister M. Venkaiah Naidu said the issue should be discussed, but the prime minister should not be dragged into this.
Speaker Sumitra Mahajan condemned the comments, but said the issue can be discussed later.
The Congress members, however, started shouting slogans which disrupted the question hour following which the speaker adjourned the house for 20 minutes.
PM Modi looks forward to productive Parliament session
As the second part of Budget session got underway, Prime Minister Narendra Modi today appreciated the cooperation of all parties in the first half and hoped that the same would continue.
"The first part of the Budget session witnessed unprecedented outcomes. Even in terms of productivity, it was successful after many years because of the cooperation of all parties. I thank all parties for that," he told reporters outside Parliament.
Listen to the audio here: Mayhem in Lok Sabha over 'Land Bill' & 'Giriraj's comment'
"I am confident that in the second part also the level of discussions will keep rising. There would be discussion as per the hopes and expectations of the common people and with the cooperation of parties good decisions would be taken and we will take steps towards meeting the expectations of the people," he said.
Earlier this morning, the Prime Minister had tweeted that he looked forward to a "productive session, filled with constructive debates on several issues".
(With inputs from agencies)