Well-off businessman Abbas was elected as chairman last weekend after beating off competitions from Mansour and Kamal Darwish.
Mansour, who's renowned for his fiery statements, alleged the elections were rigged by 'Abbas-favoring officials'.
"This is definitely the worst Egyptian election ever," the former People's Assembly member told a press conference on Tuesday.
"Abbas won because of his dishonestly orchestrated campaign and close relationship with (National Sports Council chief) Hassan Sakr."
However, Mansour assured that he had no intention to appeal against Abbas' appointment.
"As I promised, I'm not going to appeal for a re-election," he said.
"But what has been written in the press about how Abbas achieved a clean sweep urged me to reveal these facts.
"Some underhand efforts by several Abbas-favoring officials were the reasons why he won."