Sports minister Dayasiri Jayasekara told Sri Lanka's players they were too fat after they failed to reach the Champions Trophy semi-finals.
Malinga, 33, responded by saying he did not care for criticism "from those who are simply warming chairs".
"What does a monkey know about a parrot's nesting hollow?" he added.
"This is like a monkey getting into a parrot's nest and talking about it."
Jayasekara said Malinga's comments were a breach of his contract with the country's cricket board.
"In my criticism of the appalling fitness levels of our players, I did not name Malinga, but he has chosen to put the hat on and attack me publicly," he added.
Malinga won the Indian Premier League title with the Mumbai Indians before the Champions Trophy, and missed Sri Lanka's high-altitude training.
Jayasekara spoke out after Pakistan skipper Sarfraz Ahmed was dropped twice off the bowling of Malinga as Sri Lanka failed to progress from their Champions Trophy group.
"The typical body fat amounts for a cricketer should be about 16%, but most of our players have over 25%," he said.
"I want the results analysed and in future no-one will be included in a national squad if they are above 16%."
Fast bowler Lasith Malinga is being investigated by Sri Lanka's cricket board after comparing the country's sports minister to a monkey.