In general, no. Linux requires megabytes of RAM (actual writable RAM), which is much more than is present on microcontrollers. Also a memory management unit (MMU), also lacking on most microcontrollers. (although, people have apparently run versions of BSD unix on PIC32 microcontrollers with as little as 128k of internal RAM. The PIC32 has a primitive user/system MMU, and BSD Unix DID run on PDP11s at one time...)
Also, it will be strongly dependent on what you mean by "linux"; there are a great many "appliances" that run linux on little more than a SoC Microprocessor (essentially a microcontroller with no memory) and some external memories. Buy those are things like routers, with no display or UI to speak of. If your idea of "linux" is something that runs firefox and plays flash games on an HD display on facebook, that would be a much bigger system.