Is there any way to run web server site on lwip library without conversion html to hex, but directly using html files?
There should be no issues with doing that. The main issue is how the pages are stored. If you want them included in the binary, then they have to be converted and organised in a way that they can be stored in program memory and retrieved by the software. If your want to read them off of an external flash memory of some sort (e.g. Micro SD card) then you will have to implement either a custom filesystem or a filesystem your computer understand as well, such as FAT. Then all you need to do is connect lwIP to the filesystem access code so it will load the requested file. I believe there is a bit of demo code from TI for their stellaris controllers that serves pages off of a micro SD card with lwIP.