And they shouldn't be required to do so. Most foods today are GM in one way or another. It will just become a sign for consumers to stay away from those products when there's really no good reason to do so.
It's the flipside of the oft proposed "let them vote with their dollars" solution that's not often discussed: in order to make informed decisions people need (easy, plain language) access to honest, total information. If the problem of information is solved today in a non-utopian free-market, the move to one becomes less dangerous. If we shift to one from the information desert of today the consumer is just screwed for eternity.
That is, it's one problem of many with that solution; the hardest, as I see it, is proving that such a thing as a rational, informed consumer is not a myth.
If you are allowed to put completely irrelevant or misleading labels, companies could "distinguish" their brands with other nonsense:
"Biotin free!" "No added Niacin!" "No synthetic Pyridoxine!"
Does any consumer realize that these are in fact not things you should be avoiding? Of course not, and they shouldn't have to. So if you start letting companies label food as GMO free, most people will be misled because they expect the labeling to be because of a scientific reason the food is worth avoiding--when there is of course absolutely no scientific basis to fear GMO foods.
In fact, I explicitly made mention of an alternate reason for wanting to avoid GM foods is because I find their business(IP) practices vile. That's one of the textbook examples of "voting with your dollar" - don't like the way a company does business? Don't buy them.
If people want to know how their food was produced and who was involved I think they should have the ability to get that information, certainly much easier than they can today, so that they can exercise choice.
It doesn't mean that they have to follow "x-free" labeling to get it. A footnote or adjoining placard of "this fruit produced with seeds licensed from Monsanto" would do just fine.
Well, they are allowed to here in Australia. For example, we have milk marketed as "permeate-free" despite permeate being a natural milk product.
I think it's scientific to observe a single point of failure in a distribution network that everyone depends on due to a monopoly and try to mitigate it.
In lieu of warnings that you think are unfair, I think with the advent of smartphones and ubiquitous internet - it should not be too hard to have a barcode like scanner to show the origins of all the ingredients for mass produced food.
There is a "scientific" cause to fear GMO's. Consider contamination of GMO species. Or the fact that some fast-growing GMO wheat has the consequence of having little nutrients.