"Mobile se agar kisiko aapattee hai to ... mein khed vyakt karta hoon (If someone has objections to (my) mobile, then I express my regret)," Mr Thakur said amid loud protests by opposition members, asserting that he had not posted the video online.
On Monday, six Congress members were suspended for five days after they threw papers in the Lok Sabha. Mr Thakur was accused of filming the members, who had crowded around the Speaker's chair.
The Congress and other opposition parties had alleged that Mr Thakur had been let off too easily and insisted on his apology.
"If Anurag you have taken a video, please do apologise to the house," Speaker Sumitra Mahajan said, explaining that nothing had come to her notice.
After his statement regretting his action, Ms Mahajan said: "I am warning you... Dobara galti nahi honi chahiye (the mistake should not be repeated)."
Aam Aadmi Party member Bhagwant Mann, who was barred from parliament for months after he live-streamed on Facebook a video while entering parliament past layers of security, had written to the Speaker.
Congress leader Mallikarjun Kharge questioned why action was not taken against Mr Thakur but Mr Mann was suspended for two sessions.