Why do Americans call the music from Scotland and Ireland Celtic music when it has nothing to do with the Celtic languages at all?

Americans (and others) call music from Scotland and Ireland "Celtic music" not because it’s directly tied to the Celtic languages in a linguistic sense, but because it’s rooted in the cultural and historical traditions of the Celtic peoples—tribes and societies identified by archaeologists, historians, and later romanticized in popular culture. The term "Celtic" originally comes from the Greek "Keltoi," used to describe a broad group of Iron Age tribes across Europe, including those in what’s now Scotland, Ireland, Wales, Cornwall, Brittany, and the Isle of Man. These groups shared artistic styles, like intricate knotwork, and certain social structures, though their languages and music evolved differently over time.

The music itself—think fiddles, bagpipes, harps, and lively reels or haunting ballads—developed in regions where Celtic languages like Scottish Gaelic, Irish Gaelic, or Welsh were historically spoken. Even if the lyrics aren’t always in those languages today (many songs are in English or are instrumental), the melodies, rhythms, and instruments trace back to those older traditions. By the 19th and 20th centuries, Irish and Scottish immigrants brought this music to America, where it got lumped under the "Celtic" label as a catchy, broad identifier for anything evoking that misty, ancient vibe—especially during the folk revival of the 1960s and ‘70s.

You’re right to question the language link, though. The Celtic languages (Goidelic branch: Irish, Scottish Gaelic, Manx; Brythonic branch: Welsh, Breton, Cornish) aren’t a prerequisite for the music. Plenty of "Celtic" tunes have no words at all or use English, especially in modern renditions. The term’s more about a shared cultural heritage—real or imagined—than a strict linguistic tie. Critics might argue it’s a bit of a marketing gimmick, too, lumping distinct regional styles (Scottish reels vs. Irish jigs) into one exotic-sounding category for American audiences. Still, the name stuck, even if it’s a loose fit. 

"Women of Ireland", or "Mná na h-Éireann"

"Women of Ireland", or "Mná na h-Éireann" in Irish, is a beautiful song composed by Seán Ó Riada (1931--1971). The poem, on which the music is based, was written by Peadar Ó Dornín (?1704--1769), This video features the Gaelic lyrics. Below is an English translation, though there are variations (only the first two verses are sung in this version):

There's a woman in Erin who'd give me shelter and my fill of ale;
There's a woman in Ireland who'd prefer my strains to strings being played;
There's a woman in Erin and nothing would please her more
Than to see me burning or in a grave lying cold.

There's a woman in Erin who'd be mad with envy if I was kissed
By another on fair-day, they have strange ways, but I love them all;
There are women I'll always adore, battalions of women and more
And there's this sensuous beauty and she shackled to an ugly boar.

There's a woman who promised if I'd wander with her I'd find some gold
A woman in night dress with a loveliness worth more than the woman
Who vexed Ballymoyer and the plain of Tyrone;
And the only cure for my pain I'm sure is the ale-house down the road.

What's the difference between Irish Gaelic and Scottish Gaelic?

What's the difference between Irish Gaelic and Scottish Gaelic?

Not much, as it turns out:

Scottish Gaelic is basically just an older, more conservative form of Irish Gaelic. 

The Scots or Scotti were originally a Celtic tribe living in northern Ireland. Between the 6th and 10th centuries A.D., some of them migrated to northern Britain then called Caledonia, now called Scotland. They took their north Irish Gaelic dialect along with them and it later evolved into Scots Gaelic. 

Scotland was not uninhabited when they came there. The Scots found other Celts already living there in the southwest (Strathclyde) who spoke a language very similar to Welsh called "Cumbrian." 

They also found a pre-Celtic people called "Picts" or "Caledoni" living in the Highlands. Unfortunately, their language is unknown to us and survives only in 14 inscriptions - some in ogham writing and some in Roman letters - which linguists have been unable to translate. 

Nevertheless, the Scots gradually merged with the Strathclyde Celts and Picts into a single Scottish nation. Their Gaelic language eventually replaced both the Cumbrian and Pictish languages but it's not exactly known how. 

Since the language of a colony always tends to be more conservative than what it is in the mother country, Gaelic didn't change as much in Scotland as it did in Ireland. So, ironically, Scots Gaelic is actually a little closer to the Gaelic spoken by such Old Irish heroes as Brendan the Navigator, John Erugena (or Duns Scotus) and Brian Boru than modern Irish Gaelic is. 

Linguists are divided as to whether Irish and Scots Gaelic are separate languages like Norwegian and Swedish or just strongly differentiated dialects of the same language like High German and Low German. 

Nevertheless, despite some differences, the two Gaelics still seem to be largely mutually intelligible between their speakers. One Irish scholar I talked to at a book fair once said that he could still understand 95% of Scottish Gaelic whenever he read it. 

There is still a third, little known type of Gaelic called Manx (as in Manx cats) which was once spoken on the Isle of Man between northern England and Ireland. Manx was a tad bit closer to Scottish Gaelic than Irish Gaelic and simpler than both. It's speakers were mostly Roman Catholic in religion. It died out not too long ago. The last speaker of Manx died in either 1957, 1962 or 1965 depending upon whose accounts you go by.