Addressing supporters in Kabul, Abdullah Abdullah repeated claims that the election process was marred by fraud.
US Secretary of State John Kerry had earlier warned against a power grab, amid reports that Dr. Abdullah Abdullah was planning a “parallel government”.
Meanwhile, a bomb near Kabul has killed 16 people, including four NATO troops.
Ten civilians and two police officers were also reportedly killed in the attack on a clinic near Bagram, home to the largest US base in the country.
Abdullah Abdullah told the gathering in Kabul that he would never “accept a fraudulent government”.
“We are the winners of this round of elections, without any doubt,” he said, to cheers from the crowd.
However, Abdullah Abdullah stopped short of declaring a parallel government as his supporters had earlier suggested he might. Instead, he called for a unified country.
“We don’t want civil war, we don’t want a crisis,” he said.
“We want stability, national unity, not division.”
Preliminary results announced on Monday gave Ashraf Ghani 56.44% of votes in the June 14 run-off.
Abdullah Abdullah, who fell just short of an outright majority in the first round, had 43.56%.
Both of them have alleged fraud in the election. Votes are being re-checked at nearly a third of polling stations – more than 7,000.
Correspondents say recounts could significantly alter the final result, due on July 22.
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