"The call was delegated to a Minister of State because I was prioritising security and capacity at the airport on the direct advice of the Director and the Director-General overseeing the crisis response," Raab said in a statement.
The UK foreign secretary, who was vacationing in Crete, Greece, said the whole government had been working "tirelessly" over the last week to help as many people evacuate from Afghanistan as possible.
Raab’s refusal to personally speak to the Afghan minister has prompted criticism and calls to resign from the opposition.
"If Dominic Raab doesn't have the decency to resign, the prime minister must show a shred of leadership and sack him," Labour shadow foreign minister Lisa Nandy said in a statement.
On Sunday, the Taliban completed their takeover of Afghanistan by entering Kabul. Afghan President Ashraf Ghani fled the country to prevent what he described as bloodshed that would occur if militants had to fight for the city. Most countries have reduced or evacuated their diplomatic missions in Kabul.