Mr Proctor lodged a formal complaint with the broadcaster after he cut short the BBC Breakfast interview on October 5, saying he had been prevented from speaking.
He had appeared on the morning show to discuss the impact the disgraced Operation Midland had on his life, following false claims of a VIP paedophile ring.
During an exchange via videolink, the former MP levelled accusations against the Met and its commissioner Dame Cressida Dick as Munchetty attempted to speak over him to read a statement from the force.
The pair continued to talk over each other before Mr Proctor abruptly removed his earpiece and ended the conversation, saying: "I am sorry, I am not having this."
Mr Proctor, 72, said he had never walked out of an interview before in 50 years of public life.
He filed a complaint with the BBC after it had previously dismissed complaints from viewers over his interview.
Mr Proctor said BBC Breakfast editor Richard Frediani apologised to him in an email this week.
Mr Frediani said Munchetty had been faced with a "genuine editorial difficulty" when he made the allegations about what Dame Cressida had said in an interview with another broadcaster and that she was "conscious of right to reply issues".
The email apologised that Mr Proctor was upset by the exchange and the subsequent response.
In a response to Mr Frediani's email, seen by PA, Mr Proctor wrote: "Thank you for your response to my complaint. I think somewhere in your words there was an apology trying to get out."
Mr Proctor asked the corporation to "be upfront" with its apology and publish its decision in line with how other complaints are treated.
He said: "They have not gone through their normal procedure, as if a private apology will be sufficient."
He finished the response by telling Mr Frediani he was "grateful" for considering the complaint.
"Please pass on to Naga Munchetty my regards. It is not personal. It is a matter of professionalism, both hers and mine," he said.
The BBC declined to comment.