The best known version of The Water is Wide, which I assume you're referring to, was widely recorded in the early years of the folk music revival, not least by American performers -in particular Pete Seeger, I believe, who said that he had added a verse to it, having learned the rest from his sister Peggy (no word on where she had learned it). In those days, it was not always considered necessary to say exactly where you had got a song; "trad" was thought enough if you weren't going to try to copyright it for yourself, so a lot of people were probably inadvertently misinformed as to its origins. In fact, that set of the song was published (in 1904-6 in the UK, 1916 in the USA) by Cecil Sharp, who had noted it from Caroline Cox of High Ham, Somerset, in 1905; Sabine Baring Gould found related songs to related tunes in Devon and Cornwall in the 1880s. The text Sharp published was a collation of several that he had found in Somerset, but the distinctive version of the tune was, so far as I know, Mrs. Cox's. She didn't begin the song with the Water is wide verse; that came from another set, noted by Sharp from Elizabeth Mogg of Doddington, in 1904.
This version of the song is English, then, not Irish. It's worth mentioning that the DT file is unclear about precisely who its text came from (it's basically the same as the Sharp text, with a couple of verses added) but the tune given is the same, though set out differently. As to its origins: well, as you've probably already found out for yourself, it's one of those songs made up of "floating" verses that all turn up in other, often completely unrelated songs, so any attempt to trace an Ur text is pretty much doomed to failure. Certainly a large part of it turns up in Jamie Douglas (though without the Water is wide verse), but opinions differ as to whether that's where it came from or whether the ballad just incorporated an existing lyric as part of its narrative.
There are so many songs made up of floating verses of this kind, though, that it's impossible to be entirely sure which ones are genuinely related to each other, let alone whether there's any documentable "origin". You've probably already read some of the many previous discussions here, but I'll just add one link to a series of links I put in a year or so ago which may provide some interesting related information, though no definitive answers (because there aren't any):
[In the light of another year's learning, I'd say that my comments on The Butcher's Boy should be disregarded; I suggested that it was primarily an American song, but in fact it seems to have clear Scottish antecedents.]
I've said many times that the traditional musics of these islands are so closely interwoven that it's often a futile exercise to insist that one song "must be" Irish, another English or Scottish; of course it's satisfying to identify provenance when it can be done; not, I hope, for any nationalistic motive, but just because it's good to know the truth and to give credit where it's due. In the case of this particular version of this song, though, it is a matter of record that it is from England. I can't tell you whether or not it was also known in a similar form in Ireland, though that is perfectly possible, nor whether the tune is "originally" Irish, or English, or Scottish; all are possible, though, as I've said, this best-known form of it is an English one.
Ultimately, though, your audience will believe what it wants to.