Four insurgents, two wearing suicide vests, attacked the strictly neutral aid group's office in the eastern city on Wednesday, killing an Afghan guard.
Senior officials from NATO's International Security Assistance Force told Reuters they believed the Taliban launched the attack.
"The Islamic Emirate has never targeted civilians and those who, with the exception of espionage, are truly serving the people. We have never attacked them," Taliban spokesperson Qari Yousuf Ahmadi said in a statement on Friday.
The ICRC said on Friday it has suspended activities in Jalalabad because of the attack, during which Afghan police killed the attackers and rescued seven foreign staff members from the building, one of whom was slightly wounded.
The attack was the first of its kind on the ICRC in Afghanistan since it started operations in the country in 1987, and sent shockwaves through the international aid community in the country.
It came less than a week after another humanitarian group, the International Organisation for Migration, was attacked by insurgents in Kabul, killing at least three civilians and wounding four foreign aid workers.
Concerns are mounting over how the 352,000-strong Afghan security forces will cope with an intensifying insurgency once most foreign troops leave by the end of next year.
The ICRC's $90 million-a-year operations in Afghanistan are the group's biggest in the world. Some 1,800 ICRC staff work on projects ranging from providing orthopaedic limbs to the war wounded to visiting militants in Afghan jails.
Virtually all ICRC activities in neighbouring Pakistan, where a hospital in Peshawar regularly tended to wounded insurgents, were frozen last year after a British staff doctor was beheaded in Quetta in an attack blamed on the Taliban.
An ICRC water engineer, Ricardo Munguia, was shot dead in Afghanistan in 2003.
Philippe Stoll, ICRC spokesperson in Geneva, said on Friday the Jalalabad office would remain closed until "we have a thorough analysis and understanding of what has happened and who was behind it. We want to speak with everyone and we want to have (security) guarantees."
Its aid workers throughout the country have been told not to travel in Afghanistan, but their work is continuing while it reviews security.
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This is a positive development. Taliban now sense and appreciate political consequences of such attacks that's why their spokesperson has now jumped to the rescue of their stance. I am sure the attack was carried out by one of their groups which didn't realize or may be didn't really care about what may be the reaction and after all this was aimed at targeting "the foreigners". Taliban no longer have any central command or network, they are scattered across the country in different groups but under the sole umbrella of "Islamic Emirates of Afghanistan". The only united face of Taliban we see is their spokesperson.
Taliban are always right, bcoz they are angels :)
Taliban are right.