A ministry of defence spokesman said the arrest was carried out on Sunday, but declined to give further details.
The video has been shown widely on British television as well as Arabic news stations and shows a group of soldiers dragging Iraqi protesters behind a wall while a demonstration is under way, beating them with batons and kicking them.
The News of the World newspaper, which released the footage, said it had been filmed from the roof of a building by another soldier who can be heard egging on his comrades and mocking the Iraqis' pleas for mercy.
The ministry of defence said on Sunday that military police were investigating the alleged mistreatment.
British troops have been accused of abuse in the past, though not on the scale of that perpetrated by the Americans against prisoners at the Abu Ghraib jail near Baghdad.
Tony Blair, the British prime minister, said on Sunday that he took the latest allegations very seriously, and his spokesman said on Monday that it was important the public in Iraq understood the government's abhorrence of any abuse.