The 21-year-old Suzuki, whose father is Ghanaian-American and mother Japanese, said he had been racially abused on social media following Japan's shock 2-1 defeat to Iraq in Doha.
It comes after two instances of racist abuse directed at players in Italy and England during matches.
Ahead of Japan's game against Indonesia on Wednesday, coach Hajime Moriyasu said that racism was "something that just cannot happen."
"Zion is an important player for Japan and I strongly object to the people who violated his human rights and racially abused him," he said.
"If it has caused Zion stress or harm then he has the full support of the whole team.
"I want him to feel comfortable and to be able to fully concentrate on his football."
FIFA president Gianni Infantino called for worldwide stadium bans for fans and "automatic forfeits" for teams whose supporters hurl racist abuse, following the incidents in Italy and England at the weekend.
Fans aimed monkey chants at France goalkeeper Mike Maignan during AC Milan's dramatic 3-2 win at Udinese, with the game temporarily halted.
Coventry midfielder Kasey Palmer accused Sheffield Wednesday fans of doing the same to him during their Championship clash, which his team won 2-1 on Saturday.
"Football is a sport that can connect countries and people," said Moriyasu.
"It has a shared language that can bring people with different values together."
Comments still appeared to have been disabled on Suzuki's Instagram account on Tuesday.
The goalkeeper said on Monday that he accepted criticism of his performances but "would like people to stop making racist comments."
"I'm not about to let it beat me," he said.