Prime Minister Jacinda Ardern this week stopped short of announcing a government probe into the eruption that killed 16 people, saying officials would be providing advice on the issue after the upcoming holiday break, reports The New Zealand Herald.
Meanwhile, two investigations were already currently being carried out by health and safety watchdog Worksafe and the coroner.
National Party leader Bridges on Tuesday told the media that Ardern was being "unduly reticent".
"There will inevitably be those questions about whether tourists should have been on that island at this time, what the future of that tourism looks like, whether the recovery was done well," Bridges said.
"These are the sort of things New Zealanders and the international community will want to see there has been an independent inquiry on."
Ardern on Monday said that the Worksafe probe would cover questions of public safety and the obligations of tourism operators during the event, and could lead to prosecutions.
That investigation was now expected to take up to a year, she added.
Whakaari suddenly erupted when 47 people - 24 Austral, two Britons, four Germans, five New Zealanders, nine Americans, two Chinese and one Malaysian - were on hiking trip on the privately-owned island located 40 km north of Whakatane on New Zealand's North Island, Efe news reported.
Fourteen people remain hospitalized, most of them in critical condition with severe burns, at medical centres in New Zealand, while another 12 continue to fight for their lives in Australia.
The 321-metre-high Whakaari, 70 per cent of which is below sea level, is considered one of New Zealand's most active volcanoes and a tourist attraction.
Located at the southeastern end of the Pacific Ring of Fire, Whakaari last erupted in 2016 without causing fatalities.
The deadliest incident at the site occurred in 1914 when 10 miners died as a result of a landslide caused by the collapse of part of the volcano's crater.
( With inputs from IANS )