A man was charged with cruelty to animals and appeared in New Plymouth District Court early this month.
He was granted name suppression and was due back in court on June 1.
But the charge has since been withdrawn and the prosecution against the man has ceased, district prosecution manager Craig Jones said on Tuesday.
Jones said the case was not evidentially sufficient.
"There was no reasonable prospect of conviction," he said.
Any prosecutions must meet the Solicitor-General's Prosecution Guidelines, and this case did not, Jones said.
When Fuller was contacted by Open Justice, he was not aware the charge had been withdrawn.
He was angered by the news, describing it as "terrible" and "s***".
He said it had been a tough couple of weeks for his family after the death of Cash, and the update had only made it worse. He did not wish to comment further.
After the man was charged, Fuller took to social media to thank people for their support.
He also said the family was "broken" and missed Cash "like crazy".
"We are just blown away as a family for the amount of love and support the community and the whole of New Zealand has shown us. Thank you, everyone, we have definitely needed it," he posted.