The two explosions claimed by Daesh rattled the area outside the airport in the Afghan capital. The blasts killed 13 US service members and injured 18 others.
A senior Afghan health official also said 90 Afghan civilians were dead and over 100 others wounded.
Speaking from the White House on Thursday, Biden vowed to complete the evacuation and to avenge the deaths.
“Know this,” Biden said to the attackers. “We will not forgive. We will not forget. We will hunt you down and make you pay.”
“We will respond with force and precision, at our time, at the place we choose and at the moment of our choosing,” he said.
He went on to say that the withdrawal will not stop despite the attacks. “We will not be deterred by terrorists. We will not let them stop our mission.”
Biden also said he would allow additional US forces to join the evacuation efforts if necessary.
His aides said the Democratic president has not wavered on the August 31 deadline for getting everyone out of the country.
As of Thursday, the State Department said there were as many as 1,000 Americans in Afghanistan who may need help getting out.
The Biden administration has been widely blamed for the chaotic and deadly evacuation that began in earnest only after the collapse of the US-backed Afghan government and the Taliban’s takeover of the country.
Some Republicans urged the president to extend the evacuation beyond next Tuesday’s deadline.
House Republican leader Kevin McCarthy of California called for Speaker Nancy Pelosi, D-Calif., to bring the chamber back into session to consider legislation that would forbid the US withdrawal until all Americans are out.
Taliban not suspected in Kabul attacks
Elsewhere in his remarks, Biden said the Taliban have been “useful to work with” and are not suspected in the attacks.
“There is no evidence thus far that I’ve been given by our commanders in the field that there has been collusion between the Taliban and ISIS (Daesh) in carrying out what happened today.
Biden, when asked if he bears responsibility for the latest events in Afghanistan, replied, “I bear responsibility for, fundamentally, all that’s happened of late.”
“But here’s the deal,” he immediately added. “You know, as well as I do, that the former president made a deal with the Taliban.”
He was referring to a deal struck between former president Donald Trump and the Taliban under which the US would withdraw from Afghanistan in May of 2021 in exchange for US troops not being targeted.