Wow, this is complicated.

Just to show that it is even more complicated than what has been explained so far, here's a note that always appears in the Grammar Wizard at Teanglann:

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WARNING: Intermediate speaker - await further opinions, corrections and adjustments before acting on my advice.
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Apparently, that's only when the noun is given a further attribute ("Use of the Genitive" under "The Cases" in chapter 1).

Agus dhá éiscint fhicid ar riail gramadaí...

Gaeilge is also an interesting word in that the article is also often used in the genitive.

Searching for:
labhairt na Gaeilge turns up 495 results;
ag labhairt na Gaeilge 45

labhairt Gaeilge 101
ag labhairt Gaeilge 94

labhairt Ghaeilge 4
ag labhairt Ghaeilge 4

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WARNING: Intermediate speaker - await further opinions, corrections and adjustments before acting on my advice.
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Come to think of it, there probably aren’t a lot of definite proper nouns that can be used both with and without the definite article, so the case of Gaeilge may be pretty marginal from a grammatical point of view. Gaeilge itself is extremely common, of course, but I doubt there are a lot of others.

As an example of how Gaeilge can be indefinite, consider the language Gaeilge na hAlban. Anyway, this is fascinating. Thank you for asking the question!

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ЯГОН ТОҶИК НЕСТ ИНҶО???

It’s fine, but do you mean it can be used to refer to Gaelic languages in general? Gaeilge na hAlban is a definite noun (it’s a specific language), and Gaeilge by itself seems to usually mean Irish.