"I congratulate and greet Narendra Modi for his victory and becoming the PM-designate," he said in a Facebook post.
Kumar's gesture followed Modi congratulating Bihar Chief Minister Jitan Ram Manjhi.
In the post, Kumar also gave the reasons for giving up the chief minister's post.
"I did not get majdoori (wage) for my labour in the results of general elections. I thought I need to win your faith once more and resigned on moral grounds," he wrote.
"I have given up chair but not my responsibility towards Bihar. Once again I will try to win your trust and support for making a strong and prosperous Bihar and a lively India," he said.
"I have never practised politics of comfort and compromised with my ideology... I am son of a freedom fighter...my life is an open book and I have never tried to win peoples trust my marketing myself," he said.