Something I have noticed on the internet is that Americans generally refer to the Irish language as Gaelic. This always prompts a forceful response from people in Ireland correcting the use of the word Gaelic, stating that the proper English word for their language is Irish and that Gaelic refers exclusively to the language spoken in Scotland.
Use of the term Gaelic is ubiquitous in Irish American communities in the United States, most of whom trace their ancestry to late 19th or early 20th century immigrants. My grandparents and great grandparents always used the term Gaelic to refer to the Irish language and you often see the term used for Gaelic Clubs or Gaelic cultural centers in the Northeastern U.S.
I assume because of this that the original Irish speaking migrants to the U.S would have known their language as Gaelic and that sometime in the ensuing 150 years people in Ireland stopped using this term in favor of the Irish language. When and why did this happen?
We haven't! Or rather, not all of us have. Irish, Gaelic and Irish Gaelic are all rather acceptable forms in English for our language.
Historically, the issue lies mainly in language reforms in the 20th century. Following the Anglo-Irish War and the subsequent Irish Civil War, twenty-six of Ireland's thirty-two countries became a largely autonomous dominion of the British Empire known as the Irish Free State. From 1922-37, the Free State promoted the three main dialects of Irish separately in education; these dialects being Connacht, Munster and Ulster Irish (Gaeilge Chonnacht, Gaelainn an Mumhan & Gaedhlig Uladh). You'll notice that each of the different words for the language is different! We'll get back to that.
Translations of official texts from the government in this period were a sort of mix of the three dialects, with the goal of not favouring one over the other two. When the Free State became 'Éire' or the Republic of Ireland, this practice was continued. Spelling reforms came into the effect for use by the civil service in 1945, which further streamlined the language for the sake of mutual intelligibility between the dialects. Spelling reforms were followed by standardisation of grammar, which culminated in the publication of the Caighdeán Oifigiúil or Official Standard in 1958. Under Standard Irish, the language itself is referred to as 'Gaeilge', the name under which it became internationally known. Standard Irish is also what was (and is) taught in schools throughout the country, thereon out, however local dialects are still common in their respective areas.
So, for most Irish people, 'Gaeilge' was what the language was referred to in the language, and it was simply referred to as the Irish language in English. But outside of the standard, particularly in Ulster, 'Gaelic' was still a very common term. After all, the Ulster dialect has many names in Ulster, and many of them (see Gaedhlig) sound like the English word Gaelic. In keeping with 'Gaelic' being used as an approximation of Gaelic words such as Gaedhlig or Gaeilic, when referring to the Irish language, it is pronounced as it looks in English. This is in contrast to 'Scottish Gaelic' where the second word is usually said like 'gah-lick', as it refers to the Gàidhlig language.
I'm sorry that this answer is very long-winded and honestly a bit waffly, but I feel like I needed to write more than, "A lot of Irish people have called the language Gaelic for the period of their speaking English, and still do." TL;DR -- It's the perhaps more ill-informed who will try to correct you on this issue, my friend, because native Irish speakers have called it 'Gaelic' for quite a while.