Mr Abdullah made his demand ahead of a 7 November run-off, after a UN-backed panel threw out first-round votes.
He said commission chief Azizullah Lodin had "no credibility", but Mr Karzai said he had done nothing wrong. The incumbent president dismissed Mr Abdullah's allegations, saying that sudden upheaval could upset the voting process.
The row came as US President Barack Obama pledged not to "rush" a decision about whether to send extra US troops.
Mr Obama held a sixth meeting with his national security team on October 26 to discuss the future US strategy in Afghanistan.
He then spoke at a military base in Florida, telling troops he would "never hesitate" to use force if necessary. But, he added: "I will never rush the solemn decision of sending you into harm's way."
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