Kubatbek Baibolov, a leading reformer in parliament and a critic of President Kurmanbek Bakiev, fell four votes short of the 38 needed to be elected.
A new vote is expected on 2 March.
Tekebaev resigned on 27 February. He originally tendered his resignation on 13 February, but lawmakers rejected it one week later, on 20 February.
Tekebaev's offer came after calls for him to step down after a television channel on 7 February aired highly critical requires about Bakiev made by him.