The parties that did secure seats in parliament said they would support recounts only in individual precincts, based on court orders.
International observers have praised the elections as free and fair, and President Viktor Yushchenko has declared the vote Ukraine's most democratic ever.
If parliament had passed the measure, the decision would not have been binding, but would have been seen as an embarrassment for Yushchenko.
Hundreds of demonstrators gathered outside parliament today to demand a recount.
The pro-Russian Party of Regions led by Viktor Yanukovych won the most votes in the election, followed by the Yuliya Tymoshenko Bloc, and Yushchenko's own Our Ukraine bloc.
(Interfax-Ukraine, Ukrayinska pravda, AP)
The Key Players
BEHIND THE IMAGES: Click on the links below to read RFE/RL's profiles of some of the key players in Ukraine's March 26 legislative elections:
Click on the image to view a photo gallery of some of the key players in the Ukrainian elections.