In addition to the Apple Watch screens, Apple uses GT's sapphire for its camera covers and Touch ID sensors.
GT does not plan to go out of business and plans to continue operating as normal, so this might not impact the production of the Apple Watch at all.
One of the biggest surprises on the iPhone 6 was the absence of sapphire, which was long suspected to be the new screen material. Though this bankruptcy filing certainly doesn't inspire confidence, it's unlikely that Apple balked on putting sapphire on its new phones because of GT's finances; it'd make more sense that Apple realized Gorilla Glass was a better choice for phone screen, especially since Apple did choose to partner with GT for the upcoming Watch screen.
I've asked Apple if they have discussed this with GT Technologies and if they do have any alternative material backup plans and will update if I hear back. [9to5Mac]