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Apple CEO Tim Cook says his company isn't trying to wage a war with tech rivals over privacy or make it a luxury commodity, as some have suggested.

A few hours after introducing Sign In with Apple, a new device authentication option, Cook rejected the notion that the company's privacy moves were aimed at embarrassing Facebook or Google, two companies that have come under fire recently for missteps in handing their users' data.

Cook specifically noted during the feature's unveiling at WWDC 2019 on Monday the that logins from Facebook and Google are used to track people online, but with the new tool, you can use your iOS device to authenticate your login credentials.

"You know, we're not really taking a shot at anybody," Cook said in an interview with CBS News' Norah O'Donnell. "We focus on the user. And the user wants the ability to go across numerous properties on the web without being under surveillance.

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"We're moving privacy protections forward. And I actually think it's a very reasonable request for people to make," he said.

(Disclosure: CBS News is owned by CBS, the parent company of CNET.)

But Apple did have a tense moment with Google and Facebook earlier this year, when it temporarily yanked enterprise certificates -- digital signatures that both the tech giants need to run software on iPhones and iPads, shutting down internal apps that employees at Google and Facebook used to communicate with their co-workers.

Apple made the moves after it was revealed that Facebook had taken advantage of an Apple program that lets companies design apps for private corporate use, as well as test apps before they're available to you. Meanwhile, Google came under fire for an app called Screenwise Meter that invited users outside the company to earn gift cards in exchange for letting Google monitor and analyze their data.

Cook also said more users are becoming aware of privacy and the danger inherit in a lack of it.

"It's not good for our country," Cook told CBS News. "You can imagine an environment where everyone begins to think there's no privacy. And if there's no privacy, your freedom of expression just plummets. Because now you're going to be thinking about that everybody's gonna know every single thing you're doing.

"This is not good for our country, not good for democracy."

In May, Google CEO Sundar Pichai implied that Apple is turning privacy into a "luxury good."

Apple has taken high-profile actions in support of user privacy. In 2016, for instance, it refused to alter its software so that the FBI could access an iPhone 5C tied to the San Bernardino terrorist incident, arguing that the change would create a back door to all other iPhones. Last year, it unveiled features for its Safari browser that could disable tracking tools Facebook and Twitter use to keep tabs on people's browsing habits.

Representatives for Google and Facebook didn't respond to requests for comment.

More of O'Donnell's interview with Tim Cook will air on the June 4 edition of CBS Evening News.