The short answer to this question is no.
Barca have won three La Liga titles, one Spanish cup and two Champions League titles in Pep Guardiola's three years as manager at the Camp Nou, all without Fabregas.
According to the most recent FIFA Ballon d'Or, Barca already have the best three players in the world in Lionel Messi, Xavi and Andres Iniesta, and even the most biased fans of teams such as Manchester United, Chelsea, Real Madrid or anyone else cannot claim to be supporting the best team in the world at the moment, because a Fabregas-less Barcelona is the best, and has been for a while.
The old saying, "If it ain't broke, don't fix it," could be applied here, but it wouldn't tell the whole story.
From a Barcelona perspective, the purchase of Fabregas means that the Catalans club now have another world-class player. The world's best midfield has another great midfielder.
The only problem is that Fabregas was Arsenal's best midfielder, but won't be Barca's best.
He will spend time on the bench simply because he, as well as every other central midfielder in the world, can't compare to Xavi or Iniesta.
But that doesn't mean that he isn't going to be an important player for the Catalans giants.
The signing of Fabregas means that Barca will have more depth in midfield, and he is certain to have some impact from the bench considering that 31-year-old Xavi isn't getting any younger.
Plus, the move to Barca will mean that Fabregas can learn from the best midfielders in the world, and will become less of a leader and have less pressure on his shoulders, which at 24 years of age can only make him a better player.
Because of this, and the fact that he is going to be playing with some of his friends from La Masia and playing under his childhood idol, he won't be frustrated at the prospect of spending long periods of time on the bench for the first time since he was a teenager.
In terms of what Fabregas can bring to the European Champions, he is a versatile player who could not only take over in the place of Xavi or Iniesta, but could even play in an attacking midfield role, although I wouldn't expect Barcelona to alter their playing style to suit their new signing.
And although he already has over 200 appearances at Arsenal, he is a player for the future, and I'd expect him to be peaking in about three to four years. After the likes of Xavi will play a lesser role in the team, he can become the new leader of the team.
So, in the simplest terms, Barcelona don't need Fabregas, but he is still a great signing.