“If she cannot, she should do the right thing and publicly apologise,” Oppelt said in a statement.
“If the minister has any evidence… we invite her to give this to the newspaper so we can deal with it appropriately.”
Earlier, Pule told reporters a series of articles by the newspaper formed part of a highly sophisticated campaign against her.
She alleged that business people linked to the newspaper had a vested interest in trying to secure a multi-billion rand set-top-box tender and were willing to do anything to get it.
Set-top-boxes are required for the move from analogue to digital TV broadcasting.
Pule said high-profile business people and politicians tried to force her into making decisions in their favour. She detailed several stories written about her which, she claimed, sought to project her as a corrupt minister bribed with a pair of shoes. She also allegedly gave tenders to a boyfriend, meddled in tender processes, and interfered in the appointment of officials.
Pule said the newspaper did not provide a “shred of evidence”. She said Phosane Mngqibisa, the man described by the Sunday Times as her boyfriend, was in fact her “comrade”.
Pule said the matter had been taken up with the Press Ombudsman, but no legal action or police charge had been registered.
Said Oppelt: “We find it unfortunate that rather than dealing with the essence of the claims against her, she proceeds to attack the messenger of the stories.”
The numerous articles written about Pule in the last year were in the public interest and “with no other motivation in mind”, the editor said.
Pule accused several journalists of involvement in the smear campaign including Mzilikazi wa Afrika, Rob Rose, and Stephan Hofstatter.
She described Wa Afrika as a journalist with a “highly questionable” background who allegedly had an association with businessmen and companies linked to the box tender through a cellphone business venture.
Pule further claimed one of Wa Afrika’s relatives tried to secure a meeting with her and went as far as “to propose love to me”.
Oppelt said Wa Afrika was not involved in any such cellphone business.
“If the minister has evidence that these businessmen were somehow attempting to influence the Sunday Times, please can she name them and their interests in an open forum.
“Wa Afrika did not offer to suppress any… stories in exchange for information. Such an offer would be absolutely unethical.”
Oppelt said Wa Afrika came to know, a number of months after his initial stories were published, that a distant relative had been involved in a relationship with Pule.
“A relationship she now denies ever taking place. However, Wa Afrika was not aware of this relationship when researching and writing our initial stories. He obtained no information from this relative at all.”
Wa Afrika told Cape Talk radio the minister was trying to “circumcise a mosquito”.
“She is dreaming about all of this. I personally don’t have any interest in the cellphone business at all,” he told the broadcaster.
“I deny [everything]. She must produce any document or name of the company that I am involved in.”
Wa Afrika said he received information in September that there was something going on between Pule and his brother, who later admitted to it.
Wa Afrika said there was no campaign against Pule, and that “it’s got nothing to do with Stephan, nothing to do with Rob Rose”.
Oppelt rubbished claims that Hofstatter tried to plant an individual in Pule’s office to extract information, and that Rose had a close friend at a telecommunications company linked to business interests.
“He [Rose] has no such close friends at these places, only contacts and sources who he speaks to regularly in the normal course of his job,” Oppelt said.
“Minister Pule continues to avoid clarifying the key issue — the nature of her relationship with Mngqibisa.”