BRUSSELS—An emotional dispute among European Union countries over immigration broke into the open with unusually rancor Friday as national leaders accused each other of being irresponsible on one side and disrespectful on the other.

At issue was a deal reached in the early hours Friday morning to resettle throughout Europe 40,000 asylum-seekers who are now in Italy and Greece, plus 20,000 from camps outside the continent. Leaders rejected a plan to require each EU country to accept a certain number of people, opting instead...