Can you base a computer around a 32-bit microcontroller?
Absolutely! Take a look as almost every cell phone out there.  For example, the  Motorola Droid uses the TI OMAP ARM based microcontroller running Android on top of Linux.  Basically, a full blown computer operating system is on it and many other gadgets. On some of the products my client builds, they use FreeScale PowerQuicc I &amp; II 32-bit processor/controllers that can run Linux on them.  The PowerQuicc's basically have a PowerPC core along with a separate RISC processor for handling all the peripherals in a highly configurable way.  It's a glorified microcontroller.

You also have to remember that years ago computers weren't running on 32-bit processors, but more like 8-bit processors (4-bits way back then) like the Commodore 64. Then they migrated to 16-bit, 32-bit, etc.  There's really not much difference between a microcontroller versus a microprocessor of equivalent bit architecture and speed.  Microcontrollers usually lack the floating point unit, but that can be made up for with fixed point math.  For example, the original Motorola 68000 processor (16-bits) was used to power those old time Macintosh computers and then spun into a microcontroller version for many embedded electronic applications for years. 

You need to look at the role of microcontroller to understand how it is used.  Normally, when you're designing with a microcontroller, you have a highly specialized application in mind and you're trying to fit it in a smaller space than say a PC Tower. Whereas, the computer is very general purpose: crunch numbers and process user input.  When you look for a microcontroller, you're looking for one that will support the kind of interface you're building for your application.  Do you need 3 USB ports, 2 ethernet, 2 UARTs, SPI port, ATM, and CAN interface?  Some of these interfaces don't come on a typical computer such as SPI, ATM, and CAN, and a microcontroller has them built in to reduce board space.  You can look at microcontrollers as processors designed for a specific solution.