A Trump administration proposal that would deny foreigners visas or green cards if they use or are likely to use public benefits has drawn a wave of opposition.

The Inadmissibility on Public Charge Grounds proposal would add hours of legal paperwork for businesses trying to hire or retain foreign students or workers, prevent hundreds of thousands of Americans and military members from bringing their foreign spouses to the U.S., discriminate against people with disabilities, and cost U.S. taxpayers, states and cities billions of dollars in degrading public health and productivity, critics of the rule said in comments.

“People are worried...