Toffees centre-back Distin took "full responsibility" for the 2-1 loss, apologising to the club's supporters, manager David Moyes and his team-mates for the 62nd-minute blind backpass that allowed Luis Suarez to cancel out Nikica Jelavic's first-half opener.
More poor defending allowed Andy Carroll to score an 87th-minute winner as Everton's hopes of a first major trophy since 1995 went up in smoke.
Revealing he had personally apologised in the Everton dressing room after the game, Distin said: "I take full responsibility for that goal.
"I think it changed the game, so I have to accept it."
The 34-year-old's team-mates rallied around him and he added: "The guys have been amazing.
"They don't need to point at me or anything because they know I'm man enough to face my responsibilities.
"I don't need anyone to tell me what I do right or what I do wrong because I'm mature enough to know what I do right and wrong.
"That's clearly a bad mistake from me and I have to accept it, no matter the consequences.
"And, for me, the consequences are not going to the final, which is very disappointing.
"I can be only disappointed with myself.
"Right now, I need to keep my head straight.
"I'm not in a good place right now.
"I know I cost a lot of people the final and it's tough to accept.
"Personally, I just want a few days to put that behind me and re-focus.
"That's going to be very important.
"But, right now, I have to admit it is difficult because I realise where we could've been and I feel responsible for that because we are not there.
"At the moment, I don't think too much about tomorrow. I just keep having flashbacks about this bad decision from me."
Everton are a point ahead of Liverpool in the Barclays Premier League and could finish above them for the first time since 2005.
"No matter where we finish at the end of the season, we missed a final," Distin said.
"So, that won't make me feel better, no."
Distin, whose last Wembley appearance saw him taste FA Cup glory with Portsmouth four years ago, added: "I know what it is to win the cup and, today, I've obviously got the other extreme regarding the feeling.
"Only three years ago, I went on a high and now I leave the cup in the really wrong place, so it's two completely different feelings.
"That's the cup."