WASHINGTON—The White House issued a presidential proclamation on Friday night requiring many future immigrant visa applicants to show they can afford health care, a move that could make it harder for poor migrants to enter the U.S.

The action, which is set to take effect in 30 days, would require applicants, including people with ties to family members in the U.S., to show they have health insurance or prove their financial ability to pay for medical care before being issued a visa that could lead to a green card.

The...