Ozil was the target of criticism for his performances during Germany's disappointing World Cup in Russia, as well as for his meeting with Turkish President Recep Tayyip Erdogan in May.
Ozil and Ilkay Gundogan, a team mate also of Turkish descent who posed with Erdogan as well, were jeered by German fans in warm-up games before the tournament began.
Erdogan said on Tuesday the treatment of Ozil was racist and unacceptable.
The 29-year-old's decision to quit has triggered a bad-tempered public debate in Germany about its relations with its largest immigrant community.
"It's a personal decision (which) I respect," Emery told reporters in Singapore a day before Arsenal play Atletico Madrid in the pre-season International Champions Cup tournament.
Emery, who in May became the Premier League side's first new manager in 22 years after Arsene Wenger stepped down, said Ozil felt at home with Arsenal.
"We are like his home. We are like his family ... We are going to help him to feel good," Emery added.
Ozil has not commented about his decision since his arrival in Singapore on Monday with Arsenal.
Arsenal will also play Paris Saint-Germain on Saturday in the tournament.
(Reporting by Fathin Ungku; Writing by John Geddie; Editing by Peter Rutherford)
