The Gauls did speak a Celtic language, which was therefore partially similar to the Gaelic language (though Gallic and Gaelic do not belong to the same subdivision of the Celtic language family). On the other hand, the etymologies of both terms seemingly are different: Gaul can be traced back to the same Germanic root that all produced Wales and that meant stranger, and more specifically “Celt.” On the other hand, “Gael” comes from a native Irish ethnonym (something we call an endonym), Goidel (hence the modern name of a group of Celtic languages including Manx, Scottish and Irish Gaelic: Goidelic). On the other hand, it is not impossible that there was indeed a link, either because the Germanic word that produced Gaul and Wales comes from a Celtic ethnonym [1] or because there is some IE root behind that. I am no historical linguist and cannot provide more information on this possible (or rather, non-impossible) common origin; however, if we stick to the past couple of millennia, both terms are unrelated, though they strangely enough designate two people from the same linguistic group.

[1] Though the usual suspect, the name Volcæ, is not immediately similar, but once again, I am no expert of sound shifts in the proto-Celtic or Germanic languages.

(we should not think, however, that two speakers of Celtic language are actually that close from each other; here is, for instance, an outstanding post by /u/Daeres on why this is not true even for Britain. The pertinence of “Celticity” as a cultural concept becomes even weaker when adducing Continental cases).