The International Committee of the Red Cross "is a neutral, impartial and independent humanitarian organisation," Michel Masson, the head of the ICRC delegation in Ukraine, said in a statement.
"We call on everyone involved to respect the Red Cross emblem," he said, stressing that the Geneva-based organisation "depends on the full cooperation
and understanding of everyone involved in the clashes."
"Without that, it becomes very difficult for us to do our job, to respond adequately to the needs of all those affected by the violence," Masson said.
His comments came after one Swiss ICRC member and eight volunteers from the Ukrainian Red Cross were detained for several hours late in Donetsk by a group of armed men.
"One volunteer was roughed up during the incident and had to be taken to hospital," ICRC said in the statement, adding that the man had since been treated and quickly released.
Scores of people have been killed and many others injured during clashes in Ukraine's restive Russian-speaking east after pro-European protestors ousted Kiev's Moscow-backed leader several months ago.
The team was in the town to evaluate the needs of the population, the ICRC spokesman for Europe and Central Asia, David Pierre Marquet, told AFP.
"That is probably why they were freed so quickly, their mandate was very clear," he said.
The ICRC expressed gratitude to "all those responsible for resolving this incident speedily", but did not identify who had detained its members.
Earlier Saturday, however, the self-proclaimed "People's Republic of Donetsk" said its rebels had held them on suspicion they were spies.
The brief detention underscored jitters on the eve of an independence referendum in the two eastern regions of Donetsk and Lugansk.
The Red Cross, which earlier this week said it had donated medical supplies for people injured during violent clashes in Donetsk, stressed that it remained committed to "addressing the humanitarian needs arising from the situation in eastern Ukraine".