By David Hall and Benoit Morenne

New York City delayed the start of indoor dining, a key part of its reopening plan, citing concerns over the spread of coronavirus around the country, as officials across the U.S. took divergent approaches to containing the pathogen.

"We cannot go ahead at this point in time with indoor dining in New York City," Mayor Bill de Blasio said Wednesday. Indoor dining was originally scheduled to begin July 6, with the rest of the phase 3 reopening. Restaurants play a large role in the city's tourism economy, which has taken a substantial hit during the pandemic.

There were 44,766 new coronavirus cases reported in the U.S. Tuesday, according to data from Johns Hopkins University, nearing a high of more than 45,000 set over the weekend. Total confirmed cases in the U.S. exceeded 2.6 million, and more than 127,000 Americans have died.

Anthony Fauci, the nation's top infectious-disease doctor, told a Senate committee on Tuesday that new coronavirus cases "could go up to 100,000 a day" if people continue to flout advice on social distancing and face masks. "It could get very bad," Dr. Fauci said.

In Florida, which reported more than 6,000 new coronavirus cases Tuesday, Gov. Ron DeSantis said the state was "not going back" on reopening, adding that the increase in cases was caused by young people out socializing, rather than people visiting businesses.

Other states revised their quarantine policies for visitors. New York Gov. Andrew Cuomo extended quarantine requirements enacted last week to include people coming from California, Georgia, Iowa, Idaho, Louisiana, Mississippi, Nevada and Tennessee. The order is based on the prevalence of coronavirus cases and the rate of positive tests for viral infection.

Massachusetts Gov. Charlie Baker, meanwhile, relaxed his state's 14-day self-quarantine instructions by exempting visitors from seven nearby states, including New York and New Jersey. The updated travel policy goes into effect July 1.

Elsewhere, the U.S. territory of Guam reported 12 new cases in the past two days, bringing the total number there to 267. Guam's Joint Information Center said six of the new cases were U.S. military personnel.

Public health officials look at a variety of metrics, including new cases, percent positive test results, hospitalizations and deaths to gauge the level of spread in an area. Covid-19 deaths haven't yet seen the same increases as cases, positive test results and hospitalizations, in part because the U.S. is now identifying more mild and moderate cases as testing expands. But infectious disease epidemiologists caution that deaths typically lag behind other indicators in the data, as the disease often progresses over the course of several weeks in the most severe cases.

The global coronavirus death toll neared 512,000, as the number of cases world-wide crossed 10.5 million, according to Johns Hopkins.

Americans remain barred from entering the European Union for nonessential travel, even as the bloc began to open up to as many as 15 countries.

The European Commission, the EU's executive body, recommended conditions for allowing entry to people from countries ranging from Japan to Rwanda. Chief among criteria was that the health situation, measured by the number of coronavirus cases per 100,000 people in the past 14 days, should be equal to or lower than the EU average and that the overall situation in a country should be improving. The EU average in mid-June when the latest statistics were taken was 16 per 100,000. The U.S. was over 100.

The caution comes as Europe forges ahead with efforts to reopen its economy and revive growth. France's manufacturing purchasing managers index peaked at a 21-month high of 52.3 in June after the lifting of coronavirus-related restrictions on the economy, according to IHS Markit. The French government announced Wednesday it would channel an additional EUR55 million to food banks this year.

France recorded 541 new cases on Tuesday and 21 additional deaths in hospitals.

The number of people discharged from hospitals continues to outpace new admissions. There were 602 people on life support in France as of Tuesday, 17 fewer than the day before.

India, which has been struggling to contain a surge in cases, reported an additional 507 deaths, the country's highest single-day tally, bringing the total number of dead to 17,400, according to the Health Ministry. India reported 18,653 new cases, pushing the total to 585,493. Indian authorities continue to relax restrictions even as infections in the world's second-most populous country continue to rise. Starting Wednesday, interstate movement of vehicles will be permitted for essential and nonessential services.

Health authorities in South Korea said the country will conduct antibody testing every two months, and results from the first batch of tests will be released next week. "Considering the recent research from abroad, the question of whether communities will form herd immunity after a wave of Covid-19 may be wishful thinking," Kwon Jun-wook, deputy director of the Korea Centers for Disease Control and Prevention, said. South Korea reported 51 new coronavirus cases, bringing the nation's total to 12,850.

In Japan, Tokyo Disneyland reopened on Wednesday after closing for four months. Mask-clad and socially distanced Disney fans lined up to enter in the morning, but operator Oriental Land Co. limited attendance to 20% of the usual number to protect against infection.

Since a state of emergency was fully lifted on May 25, Japan has experienced a significant rise in new infections, many linked to nighttime entertainment. The government said Wednesday that 138 new cases were identified nationwide the previous day, the highest total since the emergency was lifted.

In Australia, some 300,000 people in 10 areas of Melbourne prepared to go into lockdown for four weeks in a bid to contain an outbreak in the city. Authorities said the lockdown may need to be widened if community transmissions don't slow. Victoria state, where Melbourne is located, reported 73 new cases Wednesday.

Write to David Hall at david.hall@wsj.com