However, the 36-year-old, who was taken off at half-time on Saturday, insists the players will refocus their efforts on securing European football.
He accepts, however, hopes of Champions League qualification - which remained a tantalising prospect for much of the season - are distant.
"It was a disastrous weekend for us. It was a disastrous performance and there is no hiding away from it," he said.
"Personally I was embarrassed by my own performance, it wasn't good enough as a captain and all the lads feel the same.
"We can't hide away from the fact that we have to take our medicine now, take the criticism that comes our way.
"We really felt this was going to be our year in the FA Cup but we were beaten on the day by a team that was better than us.
"Sometimes you can lose cup ties and you can be unlucky but there was no 'unlucky' about it on Saturday, we were well and truly beaten and we need to put that right.
"We have got two months left in the season and we have to bounce back and get a reaction.
"That's what the manager wants and I'm confident in the group of players that I share a dressing room with that we will do that.
"The Champions League place is in the distance: a European place, we would take that now.
"It's what we have been going for all season, making sure that next season there is European football being played at Everton. That's our aim and remains our aim.
"If we can finish in the top four then so be it, but at the moment our short-term aim is to win next Saturday against Man City and to attack the last eight or nine games of the season."
Irate fans voiced their anger at Goodison Park - which has continued over subsequent days - with a chorus of boos and half-time and the final whistle while midfielder Marouane Fellaini was jeered when he was substituted in the second half.
The Belgium international appeared to become involved in an altercation with supporters as he left the pitch and walked straight down the tunnel.
Fellaini was due to attend a signing session at the club's Everton Two store in Liverpool this afternoon but that was strategically pulled in light of Saturday's events.
A statement on Everton's website said: "Due to unforeseen circumstances this afternoon's signing session with Marouane Fellaini at Everton Two has been postponed."
The midfielder added on Twitter: "Sorry to the Everton fans, I cannot make the signing session this afternoon. Will do another one soon. Felly."
Neville has called for the players to pick themselves up and show the fight which was so lacking at the weekend when the Premier League champions come calling on Saturday.
"It's the time where we need to stand up for ourselves," he told Sky Sports News.
"We had a disappointing February, picked up our form going into March and then last week was a major setback for us.
"But this team, this club, has got major character and I fancy us to bounce back."
Everton manager David Moyes' contract runs out in the summer and the Scot has said he will not make a decision on his future until later in the season.
Critics have suggested that has not helped the situation at Everton, who dropped to seventh after Liverpool's win over Tottenham yesterday but are still level on points with their near-neighbours and seven adrift of the top four.
"That's something that only the gaffer can answer," said Neville in response to speculation about Moyes.
"He has brought all of the players in the squad to the club and we all want him to stay but that's up to the manager.
"We aren't using that as an excuse for Saturday's performance -anyone that does, that's a bit of a cop out."