WASHINGTON—President Donald Trump took aim Friday at the 2015 Iran nuclear agreement, vowing to end U.S. participation in the landmark deal unless Congress and U.S. allies are able to deliver on punitive measures targeting Tehran’s missile program, its support for regional militant groups, and any future nuclear activities.

As a first step, Mr. Trump refused to certify to Congress under a U.S. law that Iran was complying with its obligations under the nuclear agreement, charging that the country had violated the terms of the...