Actually, most of this "Linux does not have stable ABI" was maybe true 10 years ago.
In general you are going to have mostly same compatibility problems as on Windows with mostly same solution for proprietary software: package known versions of your dependencies together with your application. When done in this way, distributing of binary-only software for Linux is often even easier than on windows, because you don't have to deal with windows-specific ABI compatibility issues (like multiple instances of C runtime in same address space or interactions between SEH and C++ exceptions)
Sorry, but that's simply not true. I used to maintain the Linux port of adventure game studio (among other projects). Even when I statically linked all dependencies it often wouldn't run on one of the Linux distributions because of changes in glibc or how x libraries had been compiled on that platform.
Ask Ryan C. Gordon (icculus) what he thinks of those supposedly stable ABIs. Ask nVidia and AMD users about how thrilled they are to have to get new drivers every time the Xorg ABI changes.
I've been developing on Unix and Linux platforms for almost fifteen years now. Linux distributions (and the kernel) still don't have stable interfaces. The kind that lets users run games and applications made 10 years ago on other platforms while Linux distros some times can't use binaries from just a few years ago.
In general you are going to have mostly same compatibility problems as on Windows with mostly same solution for proprietary software: package known versions of your dependencies together with your application. When done in this way, distributing of binary-only software for Linux is often even easier than on windows, because you don't have to deal with windows-specific ABI compatibility issues (like multiple instances of C runtime in same address space or interactions between SEH and C++ exceptions)