I think that the verse was written as 'See Ya' but most people who cover Ride On sing it as 'See You' because this is how Christy pronounces the chorus.
I went to see Jimmy MacCarthy at the National Concert Hall in Dublin last week. Near the end of the gig he sang Ride On and said 'See You' in the chorus. It may have been when he wrote the song that it was 'See Ya' but everyone has been used to Christy's version over the years because Christy's cover is the most popular.