Jean-Christophe Sandoz, head of the ICRC delegation in Sudan, said Sudanese national Najmaddin Salih Musa Bishara was killed on his way home from work while traveling in a private car.
The circumstances of the shooting are unclear and police are investigating the case.
"We call on all those who bear weapons to respect people working for the ICRC or for the broader International Red Cross and Red Crescent Movement, and indeed everyone carrying out humanitarian work in Sudan," Sandoz said in a statement.
It's the second such death of a member of the ICRC staff this year. A staff member was killed by a landmine in February in South Kordafan.
In a separate statement, Human Rights Watch said Friday it was concerned by the increase in violence in Darfur. It said at least 200,000 people have been displaced by conflict so far this year.
"The suffering of Darfur's civilians at the hands of the government seems to never end," Daniel Bekele, Africa director at Human Rights Watch, said in a statement.
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