It blamed Friday's raid, in which 208 people died, on the Murle community from neighbouring South Sudan.
The government said the army had killed 60 of those who carried out the attack.
The Murle have previously been accused of carrying out cattle raids and stealing children to raise as their own.
A mother whose husband was killed and three of her children abducted by the attackers told the BBC that she has no hope of seeing her children again.
A map showing Gambella province in west Ethiopia
"I don't know if they were killed during the crossfire," Chol Malual said. "The fighting was intense and if they survived, they will be probably be killed by the Murles."
Meanwhile, additional medical personnel have been sent from the capital Addis Ababa to help treat dozens of people who were injured during the attack.
"We have treated 82 patients," a medic in the Gambella region told the BBC, "most suffering from bullet wounds to the chest, abdomen or head.
"We feel insecure here and would like the government to deploy security guards in the more dangerous areas."
Prime Minister Hailemariam Desalegn said in an address to the nation on Sunday that Ethiopia was seeking permission to cross the border for a joint military operation with South Sudan.
Mr Hailemariam said neither South Sudan's army nor rebel forces were involved in Friday's attack.
The prime minister added that "primitive and destructive forces kill people here at various times by moving from place to place".
The targets of the raid were members of the Nuer ethnic group who live in both South Sudan and Ethiopia, the AFP news agency reports.