Theoretically you could create an appliance with just a Linux kernel, Cage and Firefox (plus dependencies) and boot that with init=cage firefox. This is how most information and advertisement displays and other kiosk systems (think of stuff like the McDonalds order machines) are made.

It would be difficult to argue that this isn't an operating system though, because typical definitions are very broad. In particular, you will always need "a program that manages a computer’s resources", and you also need "the allocation of those resources among other programs" because websites are programs (in reality a browsers will liberally fork itself to take advantage of kernel process isolation, but even if it didn't the in-process threads would still qualify).

Vendors of kiosk systems probably have better chances arguing that their devices aren't general purpose than that they don't use an operating system. However if your "kiosk" system is advertising on the basis that you can do general purpose computing within a browser then your chances of arguing that you're somehow not selling a "general purpose computing device" aren't going to be very good.