New Zealand Rugby confirmed in a statement that two matches with world champions South Africa, scheduled for September 25 and October 2, had also been cancelled following advice from the Government.
Rennie, whose previous post was with Glasgow Warriors, lambasted the decision not to communicate with Rugby Australia before going public with the news the match set for later this month was in doubt.
"Bloody angry," he told reporters after the announcement. "I think it's disappointing how it has been communicated. Our boys all found out through social media.
"I thought New Zealand Rugby didn't even have the respect to consult RA about their decision so that's hugely disappointing."
A statement from NZR added that the country's women's side will no longer host Australia for matches in Christchurch on September 26 and at Eden Park on October 2.
"The decision follows advice from Government that the Springboks and Wallaroos would not be able to enter New Zealand to play the matches due to Covid-19 travel restrictions," NZR said in its statement.
New Zealand this week went into a national lockdown after a case of the Covid-19 Delta strain, which spread from Australia, was detected.
The All Blacks thrashed Rennie's men 57-22 last Saturday, having edged out their rivals 33-25 in the first encounter at the beginning of August, and were due to wrap up the series later this month.
Australia's disappointment is exacerbated by the fact they travelled to New Zealand in October, had to quarantine and then played two Bledisloe matches despite the coronavirus pandemic still ongoing.
NZR chief executive Mark Robinson insisted the decision not to travel to Perth this weekend was due to "uncertainty" and that "once the team leaves our shores, they currently can't return until November 23" with discussions taking place with SANZAAR over rescheduling the matches.
Rennie added: "I feel like there's only one of us doing what's in the best interest of the game.
"Last year, we went to New Zealand to ensure the games were played for the benefit of both countries.
"We quarantined in Wellington for two weeks. The expectation was we made a commitment to go there this year, they'd come here and they haven't honoured. It tells you more about them than us I reckon."
South Africa, who will entertain Argentina in the Rugby Championship on Saturday, have offered to host the remaining fixtures in the tournament.
Jurie Roux, chief executive at South Africa Rugby, said: "We have advised SANZAAR that we are ready and able to host the remainder of the competition in South Africa, pending our Government's approval.
"But we are now well-versed in turning on rugby Tests within the prevailing COVID restrictions and have the venues and accommodation necessary. We just need the go-ahead."
The Springboks and Pumas were due to fly to Australia on Sunday, but they have paused their flights while they await further news from SANZAAR.