About one in five primary schools lacks results and almost a quarter of English results for 14-year-olds are missing.
There are fears from schools that some test papers have been completely lost.
Mr Balls says that ministerial intervention or "commentary" on the negotiations between the Qualifications and Curriculum Authority (QCA) and the private tests contractor, ETS, "would be totally inappropriate".
Speaking to the House of Commons, Mr Balls repeated that responsibility for the problems lay with the QCA and ETS and declined to apologise.
The schools secretary said that he had been advised that marking quality was as good as in previous years so publication of the results would go ahead.
Mr Balls said the deadline for appeals against marking had been extended to 10 September, or 10 days after the start of the autumn term.
Shadow schools secretary Michael Gove accused Mr Balls of having failed to intervene when there had been warning signs of problems.
"Is it not the case that he failed to make sure contracts had been awarded properly, failed to heed warning signs and failed to act quickly to avert a fiasco which every teacher in the land could see coming?"
The Liberal Democrats' schools spokesman, David Laws, said this year's tests remained a "shambles" - and warned that assurances over the quality of marking were "complacent and premature".
The statement from Mr Balls sets out the latest figures for the return of the delayed test results for 11 and 14-year-olds - which should have been delivered to pupils a fortnight ago.
These show that there has been some catching up on the return of English results to secondary school pupils - with 77% of marks published compared with 71% last week.
For maths tests for 14-year-olds, 94% of results have been published and 93% in science.
However there has been little apparent progress in the return of missing primary school test results - with 98% of all results published, similar to the position at the end of last week.
This means that about one in five primary schools will have ended the school year without a complete set of marks for English, maths and science.
One of the primary schools still waiting for their results is St Augustine's primary school in Weymouth - in the constituency of Schools Minister Jim Knight.
Head teacher Stephen Mason says that since the first missed deadline he has been chasing missing English test results - but on Tuesday, the last day of term and the last chance to give results to the 11-year-olds in person - the results have still not been returned.
Writing to parents and pupils about his "extreme disappointment", Mr Mason said: "I can only apologise on the government's behalf for the way in which all your hard work has been treated.
"I promise to give you the English marks as soon as we receive them.
"I want to thank you for the way in which you approached these tests last May. Your conduct, preparation and hard work made your teacher and I extremely proud."
Mr Mason says that when he contacted the ETS helpline on Tuesday morning he wanted to know if the test papers had been lost.
He says he was told that they were not lost but they might have been "mislaid".
A spokesman for ETS told the BBC most of the work being carried out now for primary school test papers involved putting the marks into its results database, rather than actual marking of test scripts.