STM32F4Discovery, an HTTP server and a web page on the SD card
Yes, you can do it.

For example, when you receive request GET /file.txt HTTP/1.1 you can check if such file exists, open it, sent relevant headers and file contents.

Web server demo from LwIP will require (as stated on http://lwip.wikia.com/wiki/Sample_Web_Server) extensive rewrite, but in custom HTTP server it is possible.