WASHINGTON—The Trump administration issued a rule that would disqualify legal immigrants from permanent residency if they use certain public-assistance programs and block prospective applicants deemed likely to need them.

The rule, issued by the Department of Homeland Security on Monday, is one of the most sweeping elements of the administration’s bid to create what officials described as a tighter, more discerning U.S. immigration system. Critics of the regulation said it could hurt poor immigrants and result in widespread...