Ghani's poll rival Abdullah Abdullah has said he will reject the result of the ongoing vote count due to what he claimed was "blatant fraud" committed by Ghani, the election authorities and outgoing President Hamid Karzai.
"I respectfully ask Dr Abdullah as a national figure to respect the rule of law," Ghani told supporters in his first speech since the dispute over alleged fake votes erupted.
"We are all tired of the language of threats and unlawfulness... Our votes are clean, and we will defend each vote," he said.
"It is the people's right to elect their leader through votes. Some people have created a situation where they threaten that right," he said.
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"Nobody can challenge the election process. My demand to Dr Abdullah is to return to process," Ghani added.
The United Nations has also appealed for Abdullah to re-join the election process, which, if completed, will be Afghanistan's first democratic transfer of power.
The UN hosted talks between Abdullah and the Independent Election Commission (IEC) on Monday, but the outcome of the meeting was unclear.
But reports of the early vote count put Ghani well ahead, provoking outrage from Abdullah, and fuelling fears of instability if rival supporters take to the streets as NATO combat troops withdraw this year.
Abdullah claims that there was massive ballot-box stuffing, especially in the Ghani heartlands of the southeast where Abdullah says there were more votes cast in some areas than eligible voters.
A potential breakthrough emerged on Monday when senior IEC official Zia-ul-Haq Amarkhail, whom Abdullah had accused of fraud, resigned.
"Documents and evidence that we have show our votes were higher in all 34 provinces," Ghani said today.
Foreign diplomats have expressed alarm at the prospect of a disputed outcome that could trigger political uncertainty and civil unrest as military assistance and civilian aid declines.
The preliminary result is due on July 2 and the final result, after adjudication of complaints, is scheduled for July 22.