The 25-year-old, who owes his nickname to an apparent likeness to the animated film character Shrek, has also been left out of the trip to Lima to face Peruvian champions Deportivo San Martin in the Libertadores Cup on Thursday.
Fabbiani's weight has been one of the talking points of the season with his mother giving television interviews on her son's dietary habits.
"I've told him that even if he eats salad, if he eats three bowls of salad, it's the same as eating something else," said coach Nestor Gorosito in a radio interview.
"Because a cow eats grass and it's a cow, it eats all day and it's a cow.
"Fabbiani needs to accept responsibility and lose weight. I don't need him to go down to 92 or 93 kilos, but he has to get down to 97 and at the moment he's a hundred and a bit.
"We want him to play 90 minutes, not just half an hour."
Fabbiani, who alternates moments of brilliance with others of near-disaster and produced one of the misses of the Apertura championship last year, made an instant impact after joining River from Newell's Old Boys.
"It's my physique and I can't change it much. But I'm going to play Sunday, I'm not going to get left out of the team again and I'm going to put an end to all this," he said.
