A tearful Neymar was stretchered off during PSG's 1-0 defeat to Lyon on Sunday in Ligue 1 after sustaining a left ankle injury as a result of a tackle from Mendes.
"Thiago has received many death threats, and they shouldn't exist," Garcia said at a news conference on Tuesday.
"Today, we allow ourselves to easily insult or say horrible things.
"Thiago Mendes was sent off in Paris. I don't like the buzz around that, he attacked the ball. He was unfortunate that Neymar's foot remained under his buttock."
Neymar will undergo further tests on Wednesday to evaluate the injury but PSG coach Thomas Tuchel has not ruled out him returning to action before the winter break.
Mendes, who was shown a red card for the foul, posted a video where he apologised to Neymar and said he hoped the injury was not serious.
Mendes' foul was the sixth Neymar had suffered during Sunday's encounter.
He has sustained two fractured foot injuries since joining PSG in 2017 from Barcelona.