"We strongly disagree with the decision and the potential impact that it could have on thousands of people across northern Sudan," McDonald said.
"We've been very clear with the government. We're an independent organisation and have no links whatsoever to the ICC."
On Thursday, Oxfam said it had started to relocate its international staff to the capital, Khartoum, and had been ordered to hand over its computers and equipment to the government.
Meanwhile, thousands of al-Bashir supporters gathered in Khartoum, where the president told the crowd that the ICC was tool of the "colonialist" powers and that he rejected the warrant.
Risks and fears
The Oxfam spokesman said about 60,000 displaced people had joined their camps for assistance in the past six weeks.
"The decision that the Sudanese government has made will only hurt the Sudanese people the most. I don't think the warrant is the issue, it's how the government has chosen to respond and, unfortunately, it's taken the ICC decision out on its aid agencies.
"The agencies provide a whole range of services like food and medicine. Oxfam provides clean water and sanitation to 400,000 people in Darfur."
Save the Children, which supports 50,000 children across Sudan, said its suspension would put thousands of lives at risk.
Medecins Sans Frontieres (MSF) said that its Dutch section, which runs three sites in southern Darfur, had been expelled, leaving 200,000 patients without essential medical care.
MSF warned that the consequences of the expulsion would be "very devastating for 400,000 people, including many refugees" who depend on the aid group for humanitarian relief.
Sudanese government officials have in the past threatened to take action against Darfur-based aid groups.
Zumrawi told reporters in Geneva: "Some came for humanitarian reasons, but others did not come for humanitarian reasons."
John Lueth Ukec, Sudanese ambassador to the United Nations, accused some NGOs for spreading "lies" among the population as well as having "fabricated" reports.
"We will not tolerate organisations which violate laws," Ukec said.
Ban Ki-moon, the UN secretary-general, said on Thursday that UN aid operations in Darfur will be severely damaged if the Sudanese government follows through on its decision.
"The decision by the government of Sudan to expel 13 non-governmental organisations involved in aid operations in Darfur will, if implemented, cause irrevocable damage to humanitarian operations there," Ban said in a statement read out by his spokeswoman Michele Montas.
"These agencies are key to maintaining a lifeline to 4.7 million Sudanese people who receive aid in Darfur," the statement said, adding that Ban "appeals to the government of Sudan to urgently reconsider the above decision".
Catherine Bragg, deputy UN humanitarian chief, told reporters that the United Nations had received reports that some NGO workers were being harassed and detained by Sudanese security forces.
"We have reports of a number of international staff of NGOs who were detained for up to four hours," she said, adding that some security forces were reportedly "very intimidating, very aggressive".
Ban said in his statement that "the confiscation of equipment, money and other materials [from NGOs] is unacceptable and must end immediately."
Bragg said that there are 76 NGOs working in Darfur alongside the UN staff.
The United States and other countries also urged Sudan to reconsider its expulsion decision.
Gordon Duguid, the US state department's acting deputy spokesman, told reporters: "A number of countries are trying to convince the Sudanese government to reconsider this action."
"The United States is one of them."