David Clark described himself as an "idiot" after he broke the coronavirus social-distancing measures put in place in March, early on in the country's COVID-19 lockdown.
He drove around 12 miles to the coast from his home in Dunedin on New Zealand's South Island to take a walk with his family.
Mr Clark had earlier admitted driving to a trail at a park near his home to go mountain biking.
It comes as Scotland's chief medical officer stood down on Sunday after she was caught making trips to her second home - flouting her own social distancing measures.
"At a time when we are asking New Zealanders to make historic sacrifices I've let the team down. I've been an idiot, and I understand why people will be angry with me."
Prime Minister Jacinda Ardern admitted that "under normal circumstances" she would have sacked Mr Clark, but decided to keep him in his role as health minister amid the global pandemic and instead demote him to the bottom of the cabinet rankings.
He has also been stripped of his role as associate finance minister.
"We cannot afford massive disruption in the health sector or to our response," Ms Ardern said in a statement.
"For that reason, and that reason alone, Dr Clark will maintain his role.
"But he does need to pay a price - he broke the rules.
"While he maintains his health portfolio, I am stripping him of his role as associate finance minister and demoting him to the bottom of our cabinet rankings.
"I expect better, and so does New Zealand."
The country's national lockdown was extended for a second time this week.
