Alright, then explain why you think this is a false cause fallacy. GABA doesn't need to make you stupid. It decreases brain function is the areas affected, mainly executive functioning in the frontal lobe and allow the limbic system to go unchecked. Teenagers make more rash decisions than adults due to their underdeveloped frontal lobe, which does not keep their amygdala in check.
In regards to the false cause in the study : it's a transitional logic flaw. If A->B=true and B->C=true, that doesn't mean A->C=true. The media presented C->A and the uncredited study did not suggest that was true. (There is likely a very good reason why the paper was not directly cited. It's clear from the abstract that the conclusion drawn from it was very reaching.)
Note: Timothy Verstynen, Ph.D. is a psychologist.
The paper in which the media publication is loosely based on is here -> http://www.ncbi.nlm.nih.gov/pubmed/22879428
Note: The article also uses incorrect facts (Why am I not surprised?) The original article is here -> http://www.menshealth.com/weight-loss/being-fat-makes-you-stupid
The conclusion presented in the abstract is "Our results show that increased BMI is globally associated with a reduction in white matter integrity throughout the brain, elucidating a potential mechanism by which changes in physical health may influence cognitive health."
Notice it is suggesting "a potential mechanism" not stating "it is a fact that." The word "may" is also used.
The (A->B) (B->C) logic is not even consistent with the actual research that was done, (A MAY cause B) and (B MAY cause C), so therefore is it impossible to say (A CAUSES C). It's not even an established fact that A directly causes B or B directly causes C, there may be an observational correlation but it is not a guaranteed. Not to mention, there are many other causes for these conditions other then obesity.
The basics to logical fallacies are presented here for your reference. -> http://www.theskepticsguide.org/resources/logical-fallacies
The media article is suggesting that "a reduction in white matter integrity" causes one to be stupid, this may be true, but it is certainly not the only cause of stupidity. More importantly the point of the study was not to evaluate white matter integrity and intelligence, nor are the instruments of measuring intelligence defined in the study.
As with many media opinions, actual facts that have been determined in respectable studies are directly cited, but are unrelated to the premise. As I said before, nobody is disputing that being "fat" is unhealthy and these are the only studies cited.
I also think you may have not noticed that this media article is an advertisement for a product called speed shred.
I don't know about you, but I personally take what I read in advertisements with a grain of salt.
In regards to GABA, I'm not going to go into all the specifics, but it's a critical neurotransmitter that is required for normal brain function.
You can read about it here -> http://en.wikipedia.org/wiki/Gamma-Aminobutyric_acid
You can read about the myriad of conditions caused by low GABA here -> http://www.vitalityandwellness.com.au/low-gaba-levels
Considering the lengthy list of conditions caused by low GABA levels, I think it's safe to say that lower GABA levels are not associated with higher intelligence. Depression, excessive stress, and hypertension are known to occur though and are also caused by obesity, so your linkage is flawed (or at the very least incomplete/inadequate.)