I've already determined that statically linked programs can't always run between different distros.
My question here, is if it is safe to distribute a statically linked program that is distro specific.
For example, I'd have an Ubuntu version of the binary, and a Redhat version of the binary and the user just downloads the correct version.
Based on your clarifying answer, it seems reasonable to do that, although I'm not sure why it needs to be both statically-linked and distro-specific. Almost all major distributions distribute distro-specific binary versions of packages (e.g. binary rpms or debs). If you're going to go to the effort to distribute distro-specific pre-compiled binaries, why not distribute distro-specific packages that ensure that all dependencies are automatically satisfied?