A statement from Kravitz says: "I was really looking forward to seeing you all but we need everyone to stay safe, from the fans to my crew and band.
"Due to Covid-19 and government restrictions, my upcoming tour dates are regretfully cancelled. I am hoping to be back soon."
Ticketholders will now automatically receive a full refund for their purchase, the promoter says in a statement.
The promoter extended apologies to all those affected by the cancellation.
Kravitz's March show would have been the first time he had ever toured New Zealand.
His 11 studio albums have sold more than 40 million copies worldwide, and he also appeared in two of The Hunger Games films as the character Cinna.