No one else was being sought at this point over the tragedy, Pitkethley said. The incident was a "random, tragic event".
All Pitkethley was willing to say about the alleged offender was that he was a Waikato local.
Some members of McKinnon's family had arrived in New Zealand and were speaking with police, he said.
The family were traumatised, he said.
Witnesses had come forward to help police find the man, he said.
"I want to thank those in our community who contacted police with information and assisted us with our investigation," he said.
"I also want to acknowledge the hard work from the investigators involved in this case.
"It is because of their dedicated efforts and the assistance from the public that police were able to make an arrest within 24 hours."
In response to an Australian journalist's question about whether tourists should be concerned that this was a "wider issue", Pitkethley pointed out the high level of media interest suggested the shooting was a "random and unusual" event.
McKinnon's partner, Bianca Buckley, a midwife at Counties Manukau DHB, ran two kilometres on a dark, rugged road to find help after she watched in horror as her fiance was shot.