He referred to the report's finding that his comments following the breach that the website had been "deliberately and systematically hacked" were unreasonable, and admitted he could have done better.
"In hindsight, I accept that I could have described the incident more clearly and with a different emphasis," Makhlouf said in the letter.
However, he reiterated he was pleased his honesty and integrity and political neutrality had not been called into question and that the investigation found he acted in good faith.
Makhlouf is due to start his role leading the Central Bank of Ireland in September.
A reporter for the Irish Times, Fiach Kelly, said there had been public concern about Mr Makhlouf's appointment in recent weeks.
He said the opposition wanted the government to re-examine the appointment and the issue was likely to be raised again in the Irish parliament.