Query: precompiled libraries compatible cross platform?

This issue must be second nature to many, and I apologize for the lack
of challenge!

Would a commercially-available object-code library compiled with GCC be
compatible across platforms and/or GCC versions?

For example, if I get an object-code-only library to support a certain
WiFi peripheral hooked to an SPI on an ARM7, and the manufacturer claims
the library was compiled with GCC, does that naturally mean it is
compatible with both version 3 and 4 of arm-elf-gcc?

Even granted the identical version of arm-elf-gcc, is it also obvious
that the same object-code library file would be linkable on different
platforms, i.e., Windows, Linux, OS X?  Or does gcc require object-code
library files to be compiled per-platform?

Thanks for any wisdom folks have on this! --Bill