You are right. There is no arm-none-eabi-gcc file because the extension (“.exe” included) belongs to the file name. But I’m afraid if there was an arm-none-eabi-gcc binary file Linux would not be capable to execute it because the Windows binary format is different 
To run a Windows executable on Linux I usually use wine but even that does not work with all Windows programs.
I seems that you have to install a GCC-ARM package also on Linux.
