One of Abdullah's closest supporters said there was a strong likelihood that the former foreign minister would boycott the November 7 runoff.
"Unless these conditions are met, holding a transparent, just and free election is not possible," Ahmad Behzad, a parliamentarian who campaigned for Abdullah in the first round of the presidential poll, said.
"If the conditions are not met, there is a big possibility of a boycott by Dr Abdullah."
The members of the IEC have been appointed by Karzai and critics accuse the body of bias during the fraud-tainted August elections.
"He [Ludin] has left no credibility for the institution," Abdullah said.
"Another commissioner from the same commission should take his position."
Karzai rejected the demand, saying that a change so close to an election "will not be good for the welfare of the country".
"I hope Dr Abdullah will participate in the election. I will congratulate him if he wins, and I expect he will congratulate me if I win," Karzai said in a statement.
Meanwhile, Ludin has also said he would not step down.
"No, why should I resign?" he told the Reuters news agency.
"I don't know if the constitution gives [Abdullah] authority to ask of such a thing."
Abdullah also called for the suspension of cabinet ministers whom he accused of breaching rules of impartiality by campaigning for votes for Karzai, identifying the interior, education and tribal affairs ministers.
The United Nations-backed Electoral Complaints Committee discarded around a quarter of the votes from the August 20 poll because of fraud.
Some polling stations reported more than 100 per cent turnout while others saw ballots hugely favouring one candidate.
Ban Ki-moon, the UN secretary-general, has said that around 200 IEC field officers will be replaced for the second round.