The Education Secretary said the exams system will deliver “credible, strong results” for the overwhelming majority of young people.
It comes as universities and schools across England demanded clarity from ministers over how pupils will be able to appeal over the grades they receive.
The Government announced on Tuesday that A Level and GCSE students will be able to use results in valid mock exams to appeal if they are unhappy with their results.
Mr Williamson told the BBC: "I apologise to every single child right across the country for the disruption that they’ve had to suffer."
He added: "The system, for the overwhelming majority of young people, is going to deliver, you know, credible, strong results for every single one of them.
“It’s a robust system, it’s a fair system, it’s making sure that young people get the grades that they’ve worked so hard towards."
Mr Williamson continued: "What is key is giving young people the opportunity to move on to the next stage of their lives, making sure that they have the opportunity to go on to college, go to university, take an apprenticeship, go into the world of work."
"We’ve got a system that is, I believe, the fairest that we can do, but let’s not forget that we’ve been in a global pandemic, we’ve been in a situation none of us would have expected to be in."
Labour leader Sir Keir Starmer earlier said that it was a "blatant injustice" that young people could have their futures decided by their postcode as a result of the exams system.
"Pupils and parents are rightly worried that years of hard work are about to be undone because a computer has decided to mark their child down," he said.
"For too long, the Tories have considered the needs of young people as an afterthought when their needs should have been central.
"It’s a blatant injustice that thousands of hard-working young people risk having their futures decided on the basis of their postcode."
With less than 24 hours to go until students receive their calculated A-level results following the cancellation of exams, schools, colleges and universities are still unclear how the new appeals process will work and what the likely timescale and uptake will be among students.
England’s exams regulator Ofqual has said it is "working urgently" to set out how mock exam results will form the basis of an appeal, but further details will not be ready until next week.
Professor Julia Buckingham, president of Universities UK and vice-chancellor of Brunel University London, said: "This last-minute policy change presents a number of challenges for universities.
"We are seeking urgent clarification from the Department for Education (DfE) on a range of issues including the likely scale and timing of appeals."
Universities are concerned the appeals system may not give students enough time to secure a final grade ahead of the start of the term, and it could also cause issues with timetabling and accommodation.
The Ucas deadline for applicants to meet their academic offer conditions is September 7, which leaves exam boards less than four weeks to issue outcomes of appeals.
Dr Hollie Chandler, head of higher education policy at the Russell Group, which represents some of the most selective universities in the UK, said: "We are working with the DfE and Ofqual to get clarity on how the new appeals process will work and would urge them to ensure appeals are processed as quickly as possible to prevent further uncertainty for students and the sector."
Schools minister Nick Gibb insisted the Government had nothing to apologise for by acting so late in the day in England, adding it would only affect a small number of students.
He told BBC Breakfast: "There is no confusion. We have been very clear from the very beginning. We had to have a system in place to award qualifications to young people given that we had cancelled the exams.
"We apologise to nobody for finding solutions, even at the 11th hour, to stop any student being disadvantaged by this system."
Dr Mary Bousted, joint general secretary of the National Education Union (NEU), warned that Scottish students – who have been told they can use teachers’ predicted grades if they were downgraded after moderation – could still have an advantage over English students when applying to university.
She said: "English students are still more likely to be given lower grades – either by the exam board or by the mock exam.
"This fundamental difference in the Scottish and English awarding process does not create a level playing field between Scottish and English students for university entrance."
Following the cancellation of this year’s exams, teachers were told to submit the grades they thought each student would have received if they had sat the papers.
Exam boards have moderated these grades to ensure this year’s results in England, Northern Ireland and Wales are not significantly higher than previous years.
The National Union of Students (NUS) has called on England to follow Scotland in scrapping moderated grades.
Larissa Kennedy, president of the NUS, said: "The use of mock exams results risks making a mockery of the whole system, given the lack of a standard approach to mock exams and the fact they are not taken by all candidates.
"With its triple lock policy, all the Government has done is lock in inequality."