Hamdullah Fitrat posted vision of the aftermath on X and said the strike on Monday night had destroyed large sections of the hospital. He said the death toll so far stood at 400, while a further 250 people were injured.
Fitrat said rescue teams were trying to control the fire at the building and recover the bodies of the victims.
Pakistan had earlier denied that it had hit a hospital, saying its strike in Kabul and other strikes in eastern Afghanistan Monday had not hit any civilian sites.
The attack came hours after Afghan officials said the two sides exchanged fire along their common border, killing four people in Afghanistan, as the deadliest fighting between the neighbours in years entered a third week.
Taliban government spokesman Zabihullah Mujahid condemned the strike on X, saying it violated Afghanistan's territory. He said most of those killed and wounded were addicts undergoing treatment at the facility.
Pakistani Prime Minister Shehbaz Sharif's spokesman, Mosharraf Zaidi, dismissed the allegations as baseless, saying no hospital was targeted in Kabul.
Pakistan's Ministry of Information said the strikes "precisely targeted military installations and terrorist support infrastructure including technical equipment storage and ammunition storage of Afghan Taliban" and Afghanistan-based Pakistani militants in Kabul and Nangarhar, saying the facilities were being used against innocent Pakistani civilians.
It said Pakistan's targeting was "precise and carefully undertaken to ensure no collateral damage is inflicted".
The ministry said Mujahid's claim was "false and misleading" and aimed at stirring sentiment and cover what it described as "illegitimate support for cross-border terrorism."
It came hours after the UN Security Council called on Afghanistan's Taliban rulers to immediately step up efforts to combat terrorism.
Pakistan accuses Kabul of harbouring militant groups, particularly the Pakistani Taliban, which it says carry out attacks inside Pakistan.
The Security Council resolution, adopted unanimously, didn't name Pakistan but condemns "in the strongest terms all terrorist activity including terrorist attacks." The resolution also extends the UN political mission in Afghanistan, UNAMA, for three months.
Pakistan's government often accuses Afghanistan's Taliban government of providing safe haven to the Pakistani Taliban, which is designated as a terrorist organisation by the United States, as well as to outlawed Baloch separatist groups and other militants who frequently target Pakistani security forces and civilians across the country. Kabul denies the charge.
Earlier, Afghan officials said four people, including two children, were killed and 10 other people in southeastern Afghanistan were wounded in Monday's exchange of fire.
Mortar shells fired from Pakistan overnight struck villages in Khost Province and destroyed several homes, said Mustaghfar Gurbaz, a spokesperson for the provincial governor.
Islamabad has described the situation as an "open war." The cross-border clashes have included multiple Pakistani airstrikes on Afghanistan's capital, Kabul.
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