The the Scottish word for white geal the root of term Gaelic?
The the Scottish word for white "geal" the root of term Gaelic? And is there any connection to the word Gaul?
Probably not. The modern scholastic consensus is that “geal” comes from the Proto-Celtic for “bright”, “Gael”, comes from the proto-Brythonic for “savage” (because the old Britons viewed the old Gaels as barbarians), “Gaul” comes from old Frankish for “foreigner” (because they viewed both the Celts and the Romans as such), and Gallia (the Roman name for Gaul) comes from the Gaulish word for “power”. So it’s all one big coincidence.
You’re right that Gaul and Wales are related, but they come from Germanic *walhaz ‘foreigner’, itself of unclear – probably Celtic – origin. It (*Walha(land)) was borrowed into French as Waulle, Gaule (so the corruption happened the other way – when entering French) as the translation of Latin Gallia (itself unrelated; I’d speculate the change of W- into G- might have happened under the Latin influence, to make it more similarly-sounding to Gallia). See eg. here: https://en.wiktionary.org/wiki/Gaule#Etymology
The Germanic *walhaz was borrowed into Slavic as *volxъ ‘Romance-speaking foreigner’, hence modern Polish Włoch ‘Italian’, and the Slavic word was borrowed again into English as Vlach, Wallachia (dated words for Romanians and regions related to Romania)…