Muller responded to many questions about Walker before he visibly had had enough.
"Any questions about the road?", he asked slightly desperately, looking around as everyone can be heard awkwardly laughing.
"Come on, $1.5 billion four-lane highway. Fantastic."
People on social media have branded the remark as "embarrassing" while political commentator David Cormack said it was "a truly iconic moment in #nzpol."
Meanwhile, Herald political reporter Jason Walls took note of National MP Gerry Brownlee's facial expressions during this press conference.
During the press conference, Muller told reporters: "Hamish has decided not to stand again ... it's the right decision and now we look forward to identifying [a new] candidate."
"His judgment was fundamentally flawed."
Muller said it was not true to say that National had a track record of "dirty politics".
"One MP made a serious error and has paid the ultimate price," Muller said.
Muller reiterated his earlier statement that he only became aware of the privacy breach at midday on Monday.
Former National Party president Michelle Boag, who gave the patients' details to Walker, had recused herself from any active role in the party, which showed how serious the issue was, Muller said.
The leaking of that personal information was "totally inappropriate".
"Of course I apologise."
At the beginning of his announcement on South Island transport, Muller said National will build the Christchurch to Ashburton Expressway, originally announced as one of 10 "roads of national significance" by National in August 2017.
He said building the four-lane expressway between Christchurch and Ashburton would begin in National's first term with construction likely to start after 2023, at a cost of $1.5 billion.
