The fight in New York will be WBA, IBF, WBO and IBO champion Joshua's U.S. debut.
Coming face to face at a news conference amid tight security at a plush West London hotel, the usually calm Joshua was riled when Miller suggested the Briton was 'too posh', chinless and with his nose in the air.
"Come June 2 and Jarrell will be irrelevant," said the 29-year-old, who has a record of 22 undefeated professional fights, 21 of them by knockout and who told reporters he had 10 more years ahead of him.
"I don't know football, rugby, NFL, basketball or kick-boxing. I know how to knock people out and beat them up," added the 2012 Olympic super-heavyweight champion, in a dig at Miller's kick-boxing past.
Joshua said a knockout 'needs to happen' to announce his presence in the United States.
Monday's news conference came after a similar one in New York last week that ended in pushing and shoving between the two fighters.
If the physicality was missing in London, there were still plenty of barbs, delivered with a strong sense of theatre but no real menace.
"All that spirit that this boy's got in there... I'm going to strip it from him... I'm going to strip him of his soul in that ring. I'm going to reconstruct his face and his body on June 1. I'm looking forward to it," said Joshua.
"I'm going to be your surgeon."
Miller, also undefeated and with a record of 24 fights (23 wins and a draw), said he planned to stop Joshua in seven rounds.
The American, pretending to snore at one point as Joshua talked about his past from teenage streetfighter to world champion, said all he had heard since arriving in London was talk of Tyson Fury.
Fury and promoter Frank Warren this month signed a deal with Bob Arum's Top Rank Promotions, which has a multi-year contract with ESPN, that will see former world champion Fury fight a minimum of two bouts per year in the United States.
"He's not the powerful one, no more. I come to the UK and I hear Tyson Fury, Tyson Fury," said Miller. "He (Joshua) is a sucker, bro... he's not ready for me."
(Reporting by Alan Baldwin, editing by Christian Radnedge)
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