Josep Maria Bartomeu will replace Rosell for the remainder of his term in charge - which runs until 2016.
Reports in the Spanish media have claimed legal proceedings were to begin against Rosell regarding the deal to bring the Brazilian attacker to Camp Nou from Santos in the close-season.
The estimated €57 million deal has come under scrutiny following allegations of a misappropriation of funds, while Spanish newspaper El Mundo Deportivo reported on Monday to have seen documents showing the deal was worth nearer €95m.
Rosell quickly refuted those rumours and said he would be happy to testify in court but, after a special meeting of the board of directors was called to discuss the matter on Thursday, the situation appears to have come to a head, with Rosell confirming a departure from his presidential role.
He bemoaned "attacks" upon him and his family in a press conference, which was delayed by almost an hour and a half.
Rosell said: "Me and my family for some time have suffered attacks that make me wonder if it's worth being president.
"In recent days there has been unfair criticisms against me, coming from the national courts.
"I insist that everything that happened in the transfer of Neymar was correct.
"The club has always got to have an image that defends the club, there is confidentiality we have to expect, it's important.
"If we don't accept confidentiality it is damaging to the club. We don't want unfair publicity to affect the image of the club."
