Thuli Madonsela believes the ANC leadership supports her work.
Malunga believes "there is a certain decorum with regards to how we handle the institution we report to".
The newly appointed Malunga told the Mail & Guardian that there was no conflict or animosity between him and Madonsela and that theirs was a civil and professional relationship where they agreed to disagree.
His writing to Parliament was "an assertion of the right to differ".
"There's nothing wrong in having different views," he said. "We should, in fact, have different views."
Madonsela butted heads with members of Parliament's portfolio committee on justice and constitutional development on May 2 over her office's jurisdiction.
In a heated session, MPs said Madonsela was overstepping the boundaries of what she should be investigating, but she fought back saying they had no right to question what she investigated.
'Law unto ourselves'
On Monday Malunga wrote to the parliamentary portfolio committee, which conducts oversight over the office of the public protector, distancing himself from Madonsela's views.
"The spirit of my correspondence is that we must respect Parliament, of course, not suck up to it," Malunga said. "I have a view based on the law. Section 181 (5) of the Constitution. We have to account to Parliament. We can't be a law unto ourselves."
Malunga added, however, that Madonsela had the right to her own interpretation of the powers of the public protector.
In the letter, addressed to committee chairperson Lluwellyn Landers, Malunga wrote: "To the best of my knowledge, the views expressed by advocate Madonsela regarding our relationship with Parliament are her personal views and do not reflect my views or those of staff of the office of the public protector or any official policy adopted."
The reason for Madonsela's appearance was the presentation of her strategic plan.
According to the parliamentary monitoring group's recording, MPs questioned whether she was taking on more work than she could handle.
Appropriate institution
While he understood that the public protector had jurisdiction to investigate anything, ANC MP John Jeffery raised concern about whether the public protector's investigations were appropriately aligned to its resources. Jeffery questioned the investigation of a dispute between former national consumer commissioner Mamodupi Mohlala-Mulaudzi and trade and industry department head Lionel October after Mohlala-Mulaudzi had lost two cases in the labour court over the same matter. Jeffery said he doubted that the public protector was the appropriate institution for labour issues.
Madonsela disagreed, saying that she believed that no ombud anywhere else in the world would be asked to explain such issues.
The Democratic Alliance's Dene Smuts agreed that there was concern about the jurisdiction of the public protector and other Chapter 9 institutions and their roles, and about accountability to Parliament.
Lawson Naidoo of the Council for the Advancement of the South African Constitution said MPs had no right to ask Madonsela about her investigations, saying her office was independent of all arms of state.
"Like all Chapter 9 institutions, her office is independent of any political force. It is accountable to Parliament for operations, such as how and where it spends its money. Parliament has no right to ask why she is investigating the cases she is investigating," said Naidoo.
Madonsela's spokesperson, Kgalalelo Masibi, did not respond to several requests for comment.