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. 

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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.