The governor said Tuesday the article focused on a "small number of comments" relating to another bid city and did not reflect his "wider thoughts" on the campaign.
"I regrettably acknowledge, however, that some of my words might be considered inappropriate and consequently would like to offer my sincere apology," Inose said.
In the Times interview in New York, Inose said: "For the athletes, where will be the best place to be? Well, compare the two countries where they have yet to build infrastructure, very sophisticated facilities."
He was also quoted as saying, "Islamic countries, the only thing they share in common is Allah and they are fighting with each other, and they have classes."
International Olympic Committee rules prohibit bid cities from commenting on rival candidates. The remarks could lead to a reprimand or warning from the IOC.
Madrid is the third city bidding for the 2020 Games. The IOC will select the host city on Sept. 7 in Buenos Aires, Argentina.
Turkish Youth and Sports Minister Suat Kilic called Inose's comments "unfair and saddening."
"These remarks are not in accordance with the Olympic spirit," Kilic said in comments posted on Twitter on Saturday and confirmed by the sports ministry Tuesday. "Istanbul is a candidate city for 2020, but we haven't made any impairing comments about any other candidate city until now, and we won't.
"We like the Japanese people. We respect their culture, their elders and their youth. If the subject is Istanbul then there is no need to talk much."
Istanbul Gov. Huseyin Avni Mutlu on Tuesday called Inose's remarks "unfortunate."
"It is important that candidate cities keep to the spirit of the Olympic Games in their relations with each other," Mutlu said Tuesday in Istanbul.
Inose said he is "fully committed" to abiding by the IOC rules and looks forward to a "respectful and friendly rivalry with the other bid cities."
Tsunekazu Takeda, the Japanese Olympic Committee president and Tokyo bid leader, received an email from the IOC seeking confirmation of what happened, Kyodo News agency reported Tuesday.
Takeda insisted Inose was fully aware of the IOC rules.
"Although his sincere thoughts differ from the content of the story published, he acknowledges that his comments related to another bid city and religion may have conflicted with the IOC guidelines and, as a result, offered his profound apologies," Takeda said in a statement.
"We promise to reaffirm our utmost respect for the IOC guidelines," Takeda added. "We will continue to respect the IOC rules and give our all during the remainder of the bidding process."
Tokyo, which hosted the 1964 Olympics, is bidding for a second straight time after a failed attempt for the 2016 Games, which went to Rio de Janeiro. Istanbul is bidding for a fifth time for its first games.