The 38-year-old, who now works at Camp Nou in a sporting director capacity, underwent a transplant in 2012 following the discovery of a tumour in the organ.
A report in the Spanish media has suggested that Rosell went to the black market to purchase a liver for Abidal, with the matter brought up in court.
However, the claim has been denied by the Frenchman, the hospital where the transplant took place and also now by Barcelona themselves.
Abidal posted on Twitter: "My cousin Gerard donated his liver to save my life and for this I am truly grateful to him. Existing procedures and protocols were followed at all times and all the medical documentation that proves so is available."
Barcelona added in a statement: "The club are saddened by the lack of rigour in the spreading of such information about such a sensitive issue."
In a separate case, Rosell is to stand trial on charges of money laundering involving the sale of television rights for Brazil matches.