New York City will require people to show proof of vaccination for indoor activities, such as dining, gyms and events, Mayor Bill de Blasio said Tuesday.

The new policy will be phased in the coming weeks and will be implemented Aug. 16. By the week of Sept. 13, the city will begin inspections and enforcement, the mayor said.

The move is an effort to increase the number of people vaccinated, he said. “That is the key to protecting people and the key to recovery.”

“The only way to patronize these establishments indoors will be if you’re vaccinated, at least one dose,” Mr. de Blasio said at a Tuesday morning press conference announcing the new rules.

The seven-day average number of confirmed and probable daily coronavirus cases is now over 1,200 in New York City, according to figures this week. That figure was less than 250 at the start of July and peaked at more than 6,400 in early January, according to city statistics.

While Covid-19 vaccinations are highly effective at preventing hospitalizations or death from the virus, they’re not foolproof in preventing infection. This poses problems for events like the Olympics and raises broader questions about immunity in the long term. Photo: David Crigger/Associated Press The Wall Street Journal Interactive Edition

The seven-day average number of hospitalizations in the city has been hovering between 20 and 35 since the start of June, city data show. It was over 300 in January and February of this year.

Businesses will ask patrons to present their vaccination details in the form of the physical card or via a digital app, such as the Excelsior Pass.

Fifty-five percent of all New Yorkers including minors are fully vaccinated, according to city health data. About 60 percent of individuals have received at least one shot.

The fast-spreading Delta variant is of particular concern, as it is the dominant strain of Covid-19 in the U.S.

About 45% of New York City residents aren’t yet fully vaccinated, creating unease among some officials. “That is enough for this virus [the Delta variant] to spread,” said New York City Councilmember Mark Levine, who is also chair of City Council’s Committee on Health.

The program will be implemented through a mayoral executive order and a health commissioner’s order, Mr. de Blasio said.

“We know that this is what’s going to turn the tide,” Mr. de Blasio added. “If you want to participate in our society fully, you got to get vaccinated,” he said.

Mr. de Blasio said the safety of vaccinations was evident by the over 160 million Americans who have taken them and the only way for a full health and economic recovery was through vaccinations.

The program, called the “Key to NYC Pass,” is the most significant escalation in Covid-19 protocols in the city as the number of cases and hospitalizations rise, as well as the U.S.

For those who live outside of the city Mr. de Blasio said all Americans receive vaccination cards upon vaccination and the city would accept documentation as long as it was “accurate and real.”

Children under the age of 12, who cannot be vaccinated according to the Centers for Disease Control and Prevention, would likely not be excluded from the program’s outlined indoor activities, Mr. de Blasio said, adding the policy would be worked through in the coming days and weeks.

“This is something that we have to take into consideration. As with any policy of this type, there will have to be some reasonable accommodations made,” said Dr. Dave Chokshi, commissioner of the New York City Department of Health and Mental Hygiene, said of children under 12.

The move follows a program where individuals who receive a dose would get a $100 incentive to do so. About 11,000 New Yorkers claimed the incentive in just a few days, the mayor said.

Write to Talal Ansari at Talal.Ansari@wsj.com