If the code used for Arduino can be embedded into a permanent , callable memory, is there a way to create an OS for Arduino that runs on call?

An AVR-based Arduino is a poor choice for writing an "OS" in the modern sense. It's just too resource constrained. It doesn't have enough RAM, it doesn't have enough flash memory, it's too slow, it doesn't have protected memory or paging, it doesn't have video display, it doesn't have a file system, and on and on.

If you look at an Apple II, something like Apple DOS or ProDos gives you an idea of the sort of thing you can do (minus the video output.)

If you want to develop something like a smart watch an ARM processor is a much better choice. It packs a lot more performance into a small package. You might pick an ARM based Arduino, but you would be fighting against the lack of an OS, the lack of preemptive multi-tasking, etc.

As an alternative, a Raspberry Pi has just about everything you need. It doesn't have as many hardware interfacing options as an Arduino, but a LOT more processing power. Likewise, UNIX is a better choice as the basis for the OS for such a device. It is a protected memory OS that is made to run on relatively modest hardware.