In "The Dead", what was the cause of Gabriel's quarrel with his mother and who was proven right?

Gabriel quarrels with his mother over a conception of Gretta, Gabriel's wife. 

Gabriel's mother is described as a person with some ambitions for her children and some considerable pride. 

Gabriel remembers that she opposed his marriage to Gretta and that she called Gretta, in a derogatory way, "country cute.''

This is the source of the dispute between Gabriel and his mother. Though the text leaves open the possibility that Gabriel truly disagrees with his mother, it also leaves open the possibility that Gabriel is upset with his mother's epithet because he believes it to be true. 

At the end of the story it is the dead who "win". Gretta's young lover, Michael Furey, died when he was seventeen. Though years have passed, Michael still seems to own Gretta's affections as demonstrated when Gretta refuses Gabriel's advances in deferral to a feeling of mourning for Michael. In this choice, Gretta may prove Gabriel's mother correct. She does belong in the country. 

To use a common phrase, we might say that her heart has never left the country.