Since losing 2-1 to a touring England side in 2012, India have won 18 consecutive series at home, with only four defeats and seven draws in 53 Tests.
Rohit Sharma's side head into the three-Test series against the Black Caps starting on Wednesday on the back of a 2-0 sweep against Bangladesh but Gambhir believes their next opponents represent a completely different challenge.
"We know that they're a very good team. They've got some really high-quality players. They've got players who can hurt us. They've got players who can do the job for them as well," Gambhir said.
"They keep fighting and we expect them to fight as well. We respect them, but we don't fear anyone."
The opening Test in Bengaluru, which is followed by matches in Pune and Mumbai, faces the prospect of frustrating weather interruptions with rain predicted throughout the week.
The Black Caps, who began their Asian tour with a washout against Afghanistan in Greater Noida, face India on the back of a 2-0 defeat in Sri Lanka, with Tim Southee handing over the captaincy to Tom Latham after the embarrassing loss.
The tourists will be without batsman Kane Williamson in Bengaluru after the former skipper opted out of the opener due to a groin injury, while paceman Ben Sears was ruled out of the series with a knee injury and replaced by Jacob Duffy.
"We're obviously disappointed for Ben, who made a strong start to his Test career during the home summer and offers a genuine pace option," said New Zealand coach Gary Stead.
"It remains to be seen how long we'll be without him for, but we're hopeful his road to a full recovery will be a short one. It's an exciting opportunity for Jacob.
"With three Tests ahead of us, he has every chance of making his debut."
India top the World Test Championship table, five places above 2021 champions New Zealand, and are eager to lift a title that has slipped from their grasp twice but Gambhir is not looking too far ahead with a tour of Australia looming.
"The World Test Championship is in June 2025 and the Test matches in Australia begin on November 22. Right now, only New Zealand is on our mind and nothing else," he said.
"When you play international cricket, you don't think about how to prepare for the five Test matches in the Australia series or how to prepare for the World Test Championship final."
