I think that the op is asking whether or not they are "genetically"Irish.
Well sorry,but there is no such thing as an Irish race.
We(and I am Irish,living in Ireland)are as much a mongrel race as any other with Viking,Anglo-Saxon,Norman,and Celtic(whatever that means)blood.(Celts are either from the Iberian peninsula or central Europe,depends on what you believe.Maybe genetic groupings and markers will answer that question).
Take a quick gander at the names on for example the Irish rugby or soccer teams...a complete mix and match of both Irish and English sounding surnames.Surnames don't define heritage.