Kumar, JD(U) leader, is heading a coalition government in the state with the BJP as its junior partner.
"Congress should respect the federal structure of the country and also respect regional parties," he said.
Kumar congratulated Samajwadi Party leader Akhilesh Yadav on being elected as the chief minister of Uttar Pradesh.
Wishing him success, Kumar told reporters he has full blessing for the Samajwadi Party leader, who will be able to stabilise the law and order situation in the state.
"I wish he lives upto expectations of the people," he said and also congratulated Akali Dal leader Prakash Singh Badal for his victory in Punjab.
Citing the budget session of Bihar assembly, he said he was unable to attend the swearing-in function of Badal in Chandigarh. The chief minister also condemned the killing of IPS officer Narendra Kumar Singh in Madhya Pradesh.