Does your microcontroller have an external flash? Does it have a file system on that flash? Does that make it need an OS to support the file system?
Does it have enough memory to support allocation to different processes? Does that make it need an OS? And so on.
To your specific question: I'm not sure it goes that direction, because a full OS depends on the presence of certain features on the chip. I can think of some examples of it going the other way, though - of an SoC where you don't need or want a full OS.
And it of course depends what one means by "a OS." But, generally, if you are running multiple tasks that might depend on shared resources, you might want an OS---after all, an OS is just something that mediates shared resources among different applications.
You might prefer to use a microcontroller because of power constraints, security (generally easier to mitigate physical attacks and side channels in simpler hardware), or cost and you don't need more resources.