Times Newspapers Ltd were expected to ask the High Court in London to quash a Crown Court order requiring production of the emails to Essex Police.
Times lawyers also wanted Lord Justice Toulson and Mr Justice Cranston to declare that a judge sitting at Chelmsford Crown Court erred in law when he issued the order last October.
The case concerns emails between Vicky Pryce, Mr Huhne's ex-wife, and Isabel Oakeshott, the Sunday Times political editor.
It follows allegations that the Energy Secretary asked his former wife to take speeding points for him.
But today the publishers withdrew their application.
A Downing Street spokeswoman today declined to speculate on whether Mr Huhne could remain a minister if he is charged with an offence, saying only that members of the Cabinet are appointed by the Prime Minister and are subject to the ministerial code.
The code states that "ministers of the Crown are expected to behave in a way that upholds the highest standards of propriety" and makes clear that it is for the Prime Minister alone to judge whether they should remain in office.
The spokeswoman said: "The Prime Minister has confidence in the Energy Secretary."
Ben Silverstone, appearing for Times Newspapers, told the judges that the publishers had agreed to drop the application for judicial review at
the start of the hearing.
Today's development opens the way for the email evidence to be studied - and a swift decision made on whether or not to charge Mr Huhne.
It was announced late last year that the Crown Prosecution Service (CPS) was "very close" to making a decision and only today's application was delaying the process.
News reports have suggested ministers are on alert for a mini reshuffle if the case spells the end of the Energy Secretary's Cabinet career.
Today Andrew Edis QC, appearing for Essex Police and the CPS, described Times Newspapers' abandoned bid to block the production order as "misconceived".
The judges ordered that details of the case must remain confidential pending the decision on whether or not to prosecute.
Essex Police obtained a production order to seize the emails at a private hearing last October.
The newspaper then decided to seek judicial review of the police move, leading to the prosecution decision being put on hold.
The police investigation followed claims Mrs Pryce made in a Sunday Times interview that Liberal Democrat Mr Huhne had asked "someone" to take points on his behalf to avoid losing his licence.
It later emerged that the "someone" was allegedly Mrs Pryce.
Police have already obtained an affidavit she signed when she made her allegations to the newspaper, as well as a taped telephone conversation between her and Mr Huhne.
The speeding offence is alleged to have taken place in 2003 when the then-MEP was driving home from Stansted Airport after flying back from the European Parliament.
Mr Huhne, who left Ms Pryce for another woman, has denied the points allegations.
Mr Edis told the court that the decision whether to charge had been delayed by the Times Newspapers application.
He added: "It will now be possible to move the proceedings on towards
a charging decision. I cannot say what that decision will be."
The criminal investigation was still on foot, with more interviews to take place, and it was conceivable that there could be charges, with criminal proceedings to follow.
Mr Edis argued it was important to restrict public access to documents held by the court.
He argued they contained material which could be used in interviews or become evidence at a trial, or fall into the category of "undisclosable" material.
He told the judges: "There have been interviews already. The question is whether future interviews will be required as a result of this production order.
"Persons not presently privy to the information who may be involved in that investigation may become aware of it in circumstances that might be undesirable".
He said the possibility of the evidence being revealed would not have arisen "if these misconceived proceedings had not been issued".
Agreeing to make a confidentiality order, Lord Justice Toulson said: "Having read all the material in the case, we can see that media publication could give rise to problems if criminal proceedings are brought as a result of the pre-trial disclosure.
"We can see that it is in the interests of criminal justice that there should be a moratorium on their publication."
The judge said the order would remain in place pending the decision on whether to charge and, if charges were brought, to the conclusion of any criminal proceedings.
But the judge stressed the court did not think the order should remain in place indefinitely.
PA
