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.

The definition of OS in those laws is ridiculously broad, you can safely assume that whatever you intend to make would fall under it.

If you are concerned about the new California law it is important to point out that the California law only applies to Operating Systems vendors and providers. It does not apply to end users.

For example, if you obtain a copy of a Canadian linux distro and install and use it on your computer in California, you aren't breaking any law.

What you’re talking about would be called running a browser on “bare metal.” The OS is typically on charge of resource management between the various tasks. Access to the processor, storage, screen, input devices, sound, network. The os is a layer that mediates these devices. On bare metal you have to do ALL of that.

I’ve seen some interest in bare metal web servers in the past which some believe to be more secure. But I don’t think I’ve seen browsers on bare metal. There’s so much browsers need to do anymore. But anyways, bare metal would be the search terms you want to start using.

There is no hard definition within the laws so this is all speculation. This means that there is no technical answer because the question in is core is a legal one.

Your TV for example can have a browser without problems.

You can have an integrated board that runs a full Linux without you being able to touch the underlying OS and let that start a browser, too. You know those tv screens that show you traffic into it flight plans at the airport? Those are often full Linux computers set up exactly like that.

In short: we'll only know when the law is actually being tested. It's written in a way that I as layman could talk and software and even most hardware into it's definition, it's absolute bullshit...

Also remember, impossibility is no excuse for noncompliance now. In the US at least.