In-System Programming of FPGA by MCU
If you're serious about this, you need to read UG380, "Spartan-6 FPGA Configuration User Guide". (If it turns out that this isn't the correct FPGA familly, there's a similar document for every family Xilinx produces; just search for it.)

The Overview (starting on p. 15) shows that there are several ways to configure a Spartan-6, and JTAG isn't necessarily the simplest one to drive from a microcontroller. There's a "Slave Serial" mode that will interface directly with the SPI master controller on any micro that has one.