It comes after the actor said on a US television show that Mr Joyce looks "somehow inbred with a tomato".
Depp's wife Amber Heard avoided a conviction in court last month after illegally bringing two dogs into Australia.
But the couple were made to issue an unusual video apology.
The animosity started in 2015, when Mr Joyce threatened to kill Yorkshire terriers Pistol and Boo, which were illegally brought into Australia by Depp's wife Amber Heard.
