The Education Secretary said research due to be published by Public Health England (PHE) showed the chances of coronavirus spreading in schools was "exceptionally low".
He said it was crucial that all pupils were back at their desks next month, insisting: "The nation is ready for all children to be back in the classrooms once more."
The South Staffordshire MP said that in the event of a local lockdown schools would only be closed as a last resort, and that there was no need for staff and pupils to undergo routine tests for Covid-19.
He also rejected union calls for a staggered return of pupils on a rota system, insisting that safety measures were rigorous enough to allow for all children to return on a full-time basis.
He told the Star: "We had to make tough decisions about schooling back in March, but they were the right decisions so we could combat the pandemic.
"Schools have been working tremendously hard to get ready to be able to welcome children back into the classroom. Now is the right time to see children returning to school.
"As we get ready to deal with the new challenges that this country faces, the importance of having children in school and not missing out on any more time with their friends and their teachers and their opportunity to learn, is so paramount."
Referencing PHE research, he said: "We have to remain vigilant and safety is at the heart of all of our decisions, but it is clear that the impact on children is exceptionally low.
"There is mounting evidence that children and staff can return to school in a safe environment."
Mr Williamson said he had "full confidence" in the Government's test and trace system, saying capacity had increased so much that it was unnecessary to bring in routine testing for staff and pupils.
He said he hopes to never have to close schools again, and that in the event of a local lockdown, "there are a whole series of other actions we can take to tackle issues before we are in a position where we have to close schools".
He said the Government had issued a "comprehensive array of guidance" on how schools should operate, adding: "All the work we have been doing is geared towards a full time return. Youngsters want to get back into school, and teachers want to welcome them back.
"We will continue to work with schools and unions to provide extra clarity when required."
Mr Williamson the return of some year groups in the run up to the Summer holidays had increased confidence in the safety of schooling.
"I think people recognise that the loss that results from children being out of school is something that accumulates," he said.
"It is vital that all pupils return to school in a safe environment in September."
As part of the plans, Mr Williamson has ordered schools to have remote learning procedures in place to provide a "continuity of education" if they are forced to close again.