The Easter Road side were demoted from the top-flight after the former England skipper failed to half a disastrous run of form that has seen them win just six of his 28 games in charge.
They led Championship runners-up Hamilton 2-0 from the first leg of their play-off final but blew up spectacularly in the Easter Road return as they lost 2-0 on the day and 4-3 on penalties to join city rivals Hearts in the second tier.
Internet rumours before the game had suggested Butcher was ready to walk away from the Leith club, no matter what the result, while around 500 fans gathered outside the stadium after the game to protest against Butcher and the board.
But the former Inverness and Motherwell manager says he is determined to help rebuild the shattered club.
"If it's down to me I will be here," said Butcher. "There was stuff on social media that I would be quitting no matter what the result was. But that was never the case. I don't know where that came from. I want to continue, because I would like the opportunity to be able to restructure the club in the right way. I think that situation is out of my hands now."
Hibs were seventh when Butcher quit as Caley Thistle boss to replace Pat Fenlon at Easter Road but has now taken them back to the Championship after 15 consecutive years in the Premiership.
"You don't know how I'm feeling," he said. "It's been like watching a car crash the last two or three months, and you do everything you possibly can to stop it and you just can't. You just can't halt the slide and you can't halt the losses.
"Even today we've had opportunities to see the game out. Seventy-five seconds left and we're 1-0 down, and I'm thinking, 'okay, we'll take a 1-0 defeat, that means we stay up'. But we can't even do that.
"And then the penalties become a lottery. We miss the first penalty too. We are where we are, and it's not because of this game, it's because of what we haven't done over the whole piece, really."
Accepting responsibility for the situation, he added: "I'm the manager of the football club. I've had plenty of opportunities to stop the slide and to have won football games - that's my job. I haven't done that enough."
While Butcher was in despair, his Accies counterpart Alex Neil was left to celebrate a succesful end to his first full year as Hamilton's player-manager.
He said: "I am delighted, for the players and the fans. A lot of people wrote us off but I didn't feel it was a 2-0 game and we played well first half. I told them to believe in themselves and to believe they could do it and once we got the first goal it changed a lot.
"My team talk was just telling them to believe and telling them to create chances - and take them. I don't think the Hibs managers and players through it was job done - but people in general did.
"I always believed we would win. I just had a feeling. I don't know what it was. I was actually going to play myself but I had a feeling and I took myself out of the team again. I changed it about and to a man we were unbelievable. The effort we put in was incredible.
"I saw Kevin Phillips saying the best way to go up is through the play-offs and now that we've done it, I know what he meant. It's incredible."