Foreman, the head of mission in Sudan for MSF's Dutch branch, was released on bail of 4,000 dollars (3,200 euros), an MSF spokeman Aymeric Peguillan said.
A second Dutch MSF official, Vincent Hoedt, who is regional coordinator in Darfur, was released Tuesday afternoon on bail of 4,000 dollars (3,200 euros) after being arrested in Nyala in southern Darfur in the morning.
UN High Commissioner for Refugees Louise Arbour asked the Sudanese government to ensure that aid workers can do their job without fear of reprisals.
MSF said it was "angered by these unjustified charges and rejects the accusations."
Geoff Prescott, the head of MSF-Holland, said the arrests "were totally unacceptable" and "seriously handicap the capacity (of MSF) to provide humanitarian assistance" in Darfur.
He accused the Khartoum government of "punishing aid workers for doing their job to help the victims of the conflict in Darfur".
MSF said the charges were probably linked to Foreman's report, entitled "The Crushing Burden of Rape: Sexual Violence in Darfur", which said that around 500 women had been treated for rape in four and a half months in the western Sudanese province.
Most of the women told Medecins Sans Frontieres their attackers were from the pro-government militia known as Janjaweed or Sudanese soldiers.
Khartoum has denied the existence of any such abuse by its own troops.
"However, this report contained no accusation against the Sudanese government," said the group.
Some 10,000 aid workers operate in the Darfur region, where since February 2003 there has been a deadly conflict pitting ethnic minority rebels against the government and its proxies.
The fighting has left between 180,000 and 300,000 people dead and sparked one of the world's worst humanitarian crises.