The boy spent yesterday in the intensive care unit at Dunedin Hospital.
He was today transferred out of ICU, and was in a stable condition, a spokesman for Southland District Health Board said. He is expected to make a full recovery.
Mr Miller told Radio New Zealand he wanted a health and safety investigation into the incident to ensure no one else was hurt in similar circumstances.
The pile of top soil was situated on a former Mackenzie District Council reserve which is set to be sectioned off and sold, chief executive Wayne Barnett said, and would be subject to an internal council investigation. It had also been referred to WorkSafe New Zealand.
The top soil had been stock-piled on the site following some soil testing work, but the council had not been aware that children were using it to play on, he said.
"What has happened evidently is kids have been playing there and have dug themselves a bit of a cave or tunnel in the top soil and unfortunately that's collapsed on one of them," he said.
"We have a large number of stock-piles of various materials around, predominantly they're gravel-type material that just collapse so there's no ability to dig into them.
"We're undertaking an investigation at the moment to see what other materials are around and what measures we should take in light of what's happened."
There had been no security or safety measures at the site to cordon the soil mound off from the public, he said, and the land was still open to the public.
"We hadn't been specifically aware that kids were playing in there, although having said that, we're not hugely surprised."
He added: "It certainly wasn't something that we had foreseen. Hindsight's a tremendous educator really."
Mr Barnett had spoken to Mr Miller, who told him his son "appears to be okay, which is absolutely fantastic news".
"The key thing is that young Deon is okay, and we're tremendously relieved at that. Further to that, our focus is on having a look at what happened and making sure that we can avoid it happening again."
WorkSafe NZ confirmed it had been notified of the incident, and preliminary inquiries had begun.