The Watchtower Society repeatedly and consistently quotes this scripture to support their peculiar doctrine that the "Israel of God" is a separate group of anointed Christians with a 'heavenly hope' distinct from 99.9% of JW's who only have an 'earthly hope' and who are thus not included as a part of "the Israel of God" Paul refers to in Galatians 6:16. The Watchtower leaders are perpetuating a dispensationalist interpretation of Gal 6:16 to the effect that the "Israel of God" is the whole Church, but taking it even further to only mean a small sub-set of their JW church: the so-called "little flock" who go to heaven. The rest of JW's are excluded.
I have only done some cursory research but, in contrast to the Watchtower's unique and forced interpretation, it seems that most theologians view the "Israel of God" as merely a reference to the part of the early church who were natural Jews. One website sums it up as follows:
The Book of Galatians is concerned with Gentiles who were attempting to attain salvation through the law. The ones deceiving them were Judaizers, who were Jews demanding adherence to the Law of Moses. To them, a Gentile had to convert to Judaism before he qualified for salvation through Christ. In verse 15 Paul states that the important thing for salvation is faith, resulting in the new man. He then pronounces a blessing on two groups who would follow this rule of salvation through faith alone. The first group is the “them,” the Gentile Christians to and of whom he had devoted most of the epistle. The second group is the “Israel of God.” These are Jewish believers who, in contrast with the Judaizers, followed the rule of salvation by faith alone. Covenant Theologians must ignore the primary meaning of kai [the conjunction which is usually translated “and”] which separates the two groups in the verse in order to make them both the same group.
For Dispensational Israelology, the conclusion is that the Church is never called, and is not, a “spiritual Israel” or a “new Israel.” The term “Israel” is either used of the nation or the people as a whole, or of the believing remnant within. It is never used of the Church in general or of Gentile believers in particular. http://www.middletownbiblechurch.org/reformed/israelaf.htm
This article in particular has some excellent research: http://www.bible-researcher.com/gal6-16.html
This reminds me of JW's peculiar and unconvincing interpretation of the meaning of "other sheep" at John 10:16. Nearly all commentators state that the "other sheep" are merely the Samaritans and gentiles who Jesus would bring in to the flock, along with Jews, to become one united Christian flock regardless of one's nationality or race. Instead, the JW's interpret John 10:16 to be a prophesy by Jesus referring to the prospective "great crowd" who would form about 1900 years later, or more broadly, all Christians who have an earthly hope. Using similar strange reasoning but on the flip-side of the coin, JW's interpret the "Israel of God" at Galatians 6:16 to be a reference by Paul to only "anointed Christians" to the exclusion of all supposed non-anointed Christians who only have an earthly hope. But similarly JW's distort it to essentially refer to a group of anointed Christians who would only appear many centuries later, now in the form of the tiny minority of modern-day JW's who partake of the memorial emblems. This makes absolutely no sense when we remember that in fact all Christians in the first century were anointed (a point acknowledge by the WT), so why would Paul want to differentiate the "Israel of God" from all other Christians when he had absolutely no concept of the JW notion of two classes of Christians: one class with a "heavenly hope" (the 'faithful and discreet slave" class) and one class with an "earthly hope"?
So as it turns out, Galatians 6:16, which the Society quotes a lot, is just another JW 'proof-text' that when put under closer examination does not stand up. A little 'independent thinking' (very dangerous to JW's) and research reveals there there are more natural and convincing interpretations that make more sense in the broader scriptural context. Sadly, JW's never get to learn of those other interpretations.