Lord Justice Leveson has turned down a request from Culture Secretary Jeremy Hunt to bring forward his evidence session "in the interests of fairness to all", a spokesman for the inquiry has said.
Earlier on Friday Hunt pledged to hand over all his private texts and emails to his special adviser to inquiry into media standards.
A date has still not been set, but politicians will not be called until mid-May, meaning the Culture Secretary will have to wait at least a number of weeks.
"Lord Justice Leveson is of the view that, in the interests of fairness to all, the Inquiry should continue with the existing scheduling of his appearance," a spokesman said.
"Lord Justice Leveson has given a detailed explanation to the Secretary of State for his decision."
Hunt said he believed that his contacts with Adam Smith would show that he handled the BSkyB takeover bid by Rupert Murdoch's News Corp empire with "total integrity".
"I will be handing over all my private texts and emails to my special adviser to the Leveson Inquiry and I am confident that they will vindicate the position that I handled the BSkyB merger process with total integrity," he told reporters outside his London home.
Smith was forced to resign on Wednesday after details of his contacts with a News Corp executive were released by the Leveson Inquiry, prompting accusations of a secret "back channel" between Hunt's office and News Corp.
On Thursday evening there were renewed calls for Hunt, a Conservative minister, to resign after Lib Dem deputy leader Simon Hughes told the BBC that an inquiry was needed.
"What I cannot understand is why the matter of the ministerial code of conduct, which is to do with do you take responsibility for your special adviser, is not something the Prime Minister should immediately refer to the person who has been given the job of doing that, Sir Alex Alan," he told Question Time last night.
Nick Clegg, the deputy prime minister and Lib Dem party leader, has held the government line that people should wait to hear Hunt give evidence to the Leveson inquiry.
The affair is awkward for the Lib Dems as the party has traditionally seen itself as untainted by the phone hacking scandal as it has not been as close to News Corporation as either the Conservatives or Labour.Suggest a correction