The Scotland midfielder has recently returned to United's first-team set-up after missing the best part of two years of his football career due to colitis - a chronic bowel condition.
And the 29-year-old is convinced that United can turn around their poor league form which sees them in a lowly seventh place, seven points behind rivals Liverpool who currently occupy fourth spot.
"It's all about character now no doubt about it," Fletcher told Sky Sports. "The team has been written off, everyone has had so much to say about the situation and it's now time for the players to step up and prove people wrong and to finish the season strongly.
"It's the biggest club in the world and as soon as you are not up to a level it's like the end of the world - but every defeat is like that. We're experiencing that now and sometimes you need those moments to appreciate the success.
"Proving people wrong is always important. I think you've got to use this as a fuel. In football you've always got to look for that edge and hopefully the players can do that.
"The squad is definitely capable (of a top four finish this season). We have to go on a run, there's no doubt about that.
"As I keep saying let's finish the season strongly, get in the Champions League - we're still in the Champions League this season as well and our form has been fantastic - and let's carry that through to next season."
Fletcher insists United have been in this sort of position in the past and has backed his team-mates to demonstrate the character to put things right.
"The club has been here before many a time," Fletcher said. "I have been here personally and there are many players in that dressing room who have been there personally and you can come through it.
"It takes a bit of belief and a bit of luck but the biggest thing is to dig deep inside of yourself and I know this team is capable of it.
"I see the characters who are in the dressing room I see how much they're hurting. I'm sure we'll come through it and there will be success here in the not too distant future - no doubts about that."