The warrants accuse the two of involvement in 51 counts of war crimes and crimes against humanity including murder, torture and mass rape.
Mohamed Ali Al-Mardi, the Sudanese justice minister, said Khartoum would not hand over the suspects.
"We do not recognise the International Criminal Court ... and we will not hand over any Sudanese even from the rebel groups who take up weapons against the government," he said.
"Our position is clear and nothing has happened for us to change it."
Al-Mardi said they have investigated Haroun and found there was "not a speck of evidence" against him.
The Sudanese government says it has arrested Kosheib pending an internal investigation, but witnesses have said he has been freely traveling from one Darfur town to another under police protection.
Sudan says the ICC has no jurisdiction to try alleged crimes committed in Darfur.
However the case was referred to the court by a UN Security Council resolution which also called on Sudan to cooperate with the ICC.
The charges focus on a series of attacks on villages in Western Darfur in 2003 and 2004.
The court said there were "reasonable grounds" to conclude the pair were "criminally responsible" for war crimes and crimes against humanity.
Luis Moreno-Ocampo, the ICC's prosecutor, said: "We completed an investigation under very difficult circumstances, from outside Darfur, and without exposing any of our witnesses.
"We transformed their stories into evidence, and now the judges have confirmed the strength of that evidence."
"The judges have issued arrest warrants. As the territorial state, the government of the Sudan has a legal duty to arrest Ahmed Haroun and Ali Kosheib."
"This is the International Criminal Court's decision, and the government has to respect it," he said.
The conflict in Darfur has caused 200,000 deaths and led to two million people being displaced, according to the UN.
Sudan contests the figures, saying that only 9,000 have died.