"In a footy team there can only be one boss, that's all there is to it. I should have left because that shows they [RA] didn't trust me any more," Cheika told The Times.
"I loved Australian rugby and I thought I could do it, I believed I could get the players together and I didn't want to let the players and the supporters down.
"I tried to manage it the best way I could without being out of order. If you cause turbulence at that point, everyone feels it and I didn't need everyone to feel the turbulence in the team. You have to deal with it internally."
The 53-year-old former Waratahs No 8 quit a few days after the Wallabies' World Cup exit, revealing he had "virtually no relationship" with RA's former CEO, Kiwi administrator Raelene Castle.
Castle, who had previously led Netball New Zealand and became the first female chief executive of a club in the NRL when appointed to the role with the Bulldogs in 2013, stepped down in April following several turbulent months.