President Joe Biden said the US airlift from Kabul will continue despite the bomb blasts that killed over 90 people including 13 US service members.
"We must complete this mission and we will," he said. He also vowed to hunt down the perpetrators.
More than 100,000 people have been evacuated from Kabul, which fell to the Taliban on 15 August.
But many more Afghans have been rushing to the airport ahead of the 31 August deadline for US forces to leave.
Mr Biden vowed to complete the mission, adding "we will not be deterred by terrorists".
Thursday's attack took place at around 6pm local time.
The first targeted the Abbey Gate where US and British forces have been processing people entering the airport and was followed by gunfire.
Minutes later, the second blast occurred at a hotel that is being used by British officials to process Afghans hoping to travel to the UK.
Huge crowds had gathered in the area hoping to be allowed onto an evacuation flight - despite warnings by US and UK officials of a significant risk of suicide attacks by Isis-K, the regional branch of Islamic State.
The jihadist group later said claimed Thursday's bombings.
President Biden said: "We will not forgive. We will not forget. We will hunt you down and make you pay."
Eleven marines and a Navy medic were among those killed, said the head of US Central Command, Gen Frank McKenzie. The deaths mark the first US military casualties in Afghanistan since February 2020.
Mr Biden said the troops who had died "were heroes who were on a selfless mission to save the lives of others".
Gen Frank McKenzie said the threat from Isis-K remained high and US forces were working with Taliban to try to stop further attacks and that the Taliban had already prevented many.
But the attack is likely to significantly complicate the evacuation effort ahead of the 31 August deadline.
There are currently 5,800 US troops on the ground at Kabul airport with a further 1,000 UK troops also there.
To date, 104,000 civilians have been transferred from Afghanistan, including 66,000 from the US and 37,000 from allies and partners.
About 5,000 people are at the airport waiting at the airport and many more are still trying to get through the perimeter checkpoints.
A number of countries, including Canada, Belgium, the Netherlands and Denmark had already stopped their operations.
Turkey has announced that its troops, who had been providing security at the airport for six years, were withdrawing.