Giri won a majority of the votes in 11 of India's 17 state legislatures although the Congress Party was in power in 12.
Being a trade union leader, Giri's campaign also had the backing of the Communists and other leftist parties.
The Vice-President V V Giri announced that he would contest the presidential elections as an independent candidate.
She also went ahead and filed the nomination for Sanjiva Reddy, though she refrained from issuing a whip to Congress MPs.
Giri had edged through with a minority of Congress votes and support from a curious combination of opposition groups.