McCullum will remain in charge of the limited-overs set-up but was axed from the red-ball role he had held since 2022 after overseeing seven defeats in the last nine matches.
The New Zealander was euphemistically said to have "stood down" in an official England and Wales Cricket Board statement released on Sunday, but was quick to admit that he had been pushed.
"Yeah, I got a tap on the shoulder," he said.
"I was disappointed but at the same time I fully respect the decision. We're in a results business and fundamentally our results weren't good enough. It's time for someone else to have a go.
"We didn't always get it right. For that, I put my hand up and say we weren't able to get there."
A 2-1 defeat to New Zealand spelled the end of his Test tenure but he was already hanging by a thread in the aftermath of a chaotic 4-1 defeat in the Ashes over the winter.
That extended a poor record in the flagship contests against the other members of the so-called 'big three', losing away to Australia and India and drawing 2-2 against both on home soil.
"We just weren't quite able to win those big series, against India and Australia, the two marquee series. We just weren't able to get the results and for that I guess I can only say sorry," he said.
"I was the leader of that group. I was in charge of the team culturally, in charge of the team tactically, in charge of the team results-wise as well.
"If you don't get the results, being a results business, fundamentally you get replaced. I'm not unaccustomed to that, I've been around this game for 20-odd years and I know if you aren't getting the results, someone else needs an opportunity.
"I put my hand up for that and accept it wasn't good enough."
McCullum told BBC Sport there was a poetic element to the timing of his departure, less than two weeks since former captain Ben Stokes' dramatic Trent Bridge farewell.
"I got some nice messages from Stokesy, actually. I guess, in a romantic kind of way there's something about Stokesy and me going out together," he said.
"We started it together and we go out together, and I have no problem with that."
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