• "By June, nobody is going to be wearing masks," Dr. Scott Gottlieb told CNBC on Monday.
• The former FDA chief said he expects by next month the prevalence of Covid is going to be "sufficiently low in this country [and] we're just not going to be concerned about it."
• However, Gottlieb said people who continue to choose wearing a mask "should be supported."

Dr. Scott Gottlieb said Monday he expects more Americans to stop wearing face coverings to protect against the coronavirus in the coming weeks.

"By June, nobody is going to be wearing masks. By June, I think, the prevalence [of Covid] is going to be sufficiently low in this country [and[ we're just not going to be concerned about it," the former Food and Drug Administration commissioner said on CNBC's "Squawk Box."

Gottlieb's comments came during a back-and-forth about how the updated mask guidance from the Centers for Disease Control and Prevention impacts young children who are not yet eligible to receive a Covid vaccine. Kids aged 12 to 15 became eligible for the Pfizer-BioNTech vaccine last week.

While he's fully vaccinated, Gottlieb said he wore a mask this weekend when shopping with his young children. Gottlieb said he no longer feels at risk of getting Covid, but has no problem wearing a mask for the time being when he's with them.

"What's the downside of keeping it on?" he said. "I would probably keep a mask on my kids a little bit longer. Another week or two," added Gottlieb, who now serves on Pfizer's board.

On Thursday, the CDC said in most indoor and outdoor settings, fully vaccinated people no longer need to wear a face covering or maintain 6 feet of social distance from others. Masks are still needed on airplanes and public transportation, according to a federal rule.

States and localities also are still able to set their own policies around face coverings. New Jersey, for example, intends to keep its indoor mask requirement in place. By contrast, states such as North Carolina and Ohio relaxed their mask rules for fully vaccinated people following the CDC guidance.

Major retailers such as Walmart, Costco, Target and Starbucks have also dropped face-covering requirements for fully vaccinated customers, except in places where local rules still necessitate them.

The CDC has defended its new recommendations despite causing some confusion and concern. Critics worry that it will embolden unvaccinated people to go without a mask.

"People who are going to go maskless now were going to go maskless anyway post when these mandates lifted and still choose not to get vaccinated," said Gottlieb, who led the FDA in the Trump administration from 2017 to 2019.

Gottlieb repeated his belief that the CDC's updated guidance will incentivize at least some vaccine-hesitant Americans to get Covid shots. "I don't think it's going to be 10%, but I think we might pick up another 2% to 5% of people," he predicted.

As of Sunday, 47% of the U.S. population has received at least one Covid vaccine dose while 37% is fully vaccinated, according to CDC data.

New coronavirus infections in the U.S. remain in decline, which is why Gottlieb believes the CDC mask guidance is appropriate.

The seven-day average of new daily coronavirus cases is around 33,200, according to a CNBC analysis of Johns Hopkins University data. That's down 19% from a week ago, continuing a sharp drop that's taken place since mid-April, at which point the daily average of new cases was more than 71,000.

"I think prevalence is really collapsing around the country, and we're going to be at a point where there's very little infection and the individual risk to a person is low if you're dealing with an otherwise immunocompetent person," Gottlieb said.

At the same time, Gottlieb said people who choose to wear a mask, even when it's not required, should not be derided. For example, someone who is vaccinated but has a preexisting condition may still don a mask in certain higher-risk indoor settings, he said.

"I think it should be encouraged. I think it should be supported. People are going to have to make individual choices about their risk," Gottlieb said.

Disclosure: Scott Gottlieb is a CNBC contributor and is a member of the boards of Pfizer, genetic testing start-up Tempus, health-care tech company Aetion and biotech company Illumina. He also serves as co-chair of Norwegian Cruise Line Holdings' and Royal Caribbean's "Healthy Sail Panel."