I found this interesting about Georgia (from wiki) - The Muslims constitute from 9.9% (463,062) to 10-11% of Georgia's population. There are two major Muslim groups in Georgia. The ethnic Georgian Muslims are Sunni Hanafi and are concentrated in the Autonomous Republic of Adjara of Georgia bordering Turkey. Islam claims the second largest number of followers in Georgia, most notably among Abkhazians and Georgians from Adjara and other regions in the south. Azerbaijanis, most of whom live in Kvemo Kartli Region, and the majority of the Kist people in Pankisi Gorge also identify as Muslim.
So, it may matter what part of Georgia we are talking about, if the answer is Islam.
It would seem that the answer to why the population might be upset about gays is religion. In this case, it looks like that is not the case. 80 - 91% consider themselves to be Orthodox Eastern Christian. There are a smattering of Jews, Jehovah's Wittneses, etc. The data also suggests that it is not a practicing group of orthodox, as about only 25% of them, or close to that go to church regularly.
Based on this, I would assert that the people there just don't want western changes to come their way? Just a guess.