Mr Kumar said on Wednesday that he had to resign because of corruption probes against the leaders of his alliance partner, Rashtriya Janata Dal (RJD).
Mr Kumar has found a new partner in PM Narendra Modi's Bharatiya Janata Party (BJP), which he defeated in 2015 in a bitterly contested state election.
The RJD has accused him of "betrayal".
Former federal minister Laloo Prasad Yadav, who leads the RJD, said Mr Kumar had colluded with the BJP to tarnish his party's image.
Mr Yadav and his son Tejaswi, who was the deputy chief minister in Mr Kumar's government, are being investigated for corruption by India's Central Bureau of Investigation.
They have denied any wrongdoing.
Mr Kumar told reporters that he had a "zero tolerance" policy on corruption, and had asked his deputy to explain the charges against him to the public.
He added that Mr Yadav refused to follow his advice.
Mr Kumar's new stint also marks his return to the BJP-led National Democratic Alliance (NDA).
He had quit the NDA in 2013 citing differences with the BJP's decision to announce Mr Modi as its prime ministerial candidate for the 2014 general elections.
Nitish Kumar has taken oath as the chief minister of the northern Indian state of Bihar, a day after resigning from the post.