Veglio, his successor at the Council for Migrants and Itinerant Peoples, has held several important Vatican positions and is the former secretary of the Congregation for the Oriental Churches.
The migrants' council had on February 20 strongly criticised the adoption of a proposal by the Italian government allowing night patrols in big cities to fight crime.
But the Vatican distanced itself from the condemnation the following day, saying this was not its official position.
It was not clear if Martino's resignation was related to that incident.
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