The new health-care requirement, first reported by The Wall Street Journal, was slated to take effect Monday. It would primarily affect new immigrants looking to join family members in the U.S.—a phenomenon President Trump has called “chain migration.”

When it issued the new policy last month, the White House said it was taking the step to safeguard the health-care system for Americans by preventing immigrants from enrolling in Medicaid or going to emergency rooms with no insurance.

The White House didn’t respond to a request for comment on the judge’s order.

Immigrant advocates said the policy would effectively ban poor immigrants. Mr. Trump issued the order using the same authority as his executive order blocking citizens of several Muslim-majority nations from entering the U.S.

Mr. Trump has frequently criticized the nation’s legal immigration system, which allots most visas to family members of U.S. citizens and awards 50,000 green cards each year to foreigners in countries with low numbers of immigrants in the U.S., many of them in Africa and Asia.

Under the new policy, immigrants entering the country using either route would need to demonstrate their financial ability to purchase health care before being granted permission to move to the U.S.

The State Department issued a plan for implementing the policy on Oct. 29—days before it was set to take effect—and gave the public two days to comment rather than the usual one to three months.

One analysis of the policy’s impact, by the nonpartisan Migration Policy Institute, found that it would have blocked up to 65% of immigrants who have moved to the U.S. in recent years—about 375,000 annually.

“We’re feeling very relieved, because the proclamation, had it gone into effect, would have been the shutdown of the family-based immigration system,” said Stephen Manning, executive director of the Innovation Law Lab, one of the groups that is suing the Trump administration to halt the new policy.

The health-care requirement is part of a broader effort by President Trump to restrict immigrants’ ability to enter the U.S. In October, five separate federal courts blocked another Trump administration policy, issued by the Department of Homeland Security, that would have required many of the same applicants to demonstrate that they wouldn’t become reliant on any public benefits should they be allowed to immigrate to the U.S. That rule would have also blocked immigrants already in the U.S. from gaining permanent residence if they had used public benefits or might in the future.

Mr. Trump has also cut the number of refugees allowed into the country, to a record-low 18,000 for the current fiscal year, which began in October.

Low-income immigrants living in the country legally are already barred from using Medicaid during their first five years, but they can receive premium subsidies under the Affordable Care Act if their incomes qualify. Under this latest administration policy, purchasing health insurance on an ACA exchange using subsidies would disqualify people from living in the country legally.

The Trump administration separately repealed a mandate that most Americans obtain health insurance or pay a penalty—one of the least-popular elements of the ACA—in its tax overhaul passed in late 2017.

Write to Michelle Hackman at Michelle.Hackman@wsj.com