Docker doesn't directly do cross-platform: the Docker API is the Linux kernel userspace API. If you want to run Dockerized apps on OS X or Windows, then you're using boot2docker, which is a distribution of Linux running inside a VM.
So most of the question is whether you're happy with this as a distribution mechanism. The best you'll be able to do is something like VMware's Unity mode, which isn't bad, but isn't good either. Alternatively, you could run an X11 server on Windows (via Cygwin) and OS X (Xdarwin), but that's also going to look very non-native.
If you're not, then there's an open technical problem of how to make it look reasonable. I'd consider whether browser-based desktop apps are an option: you can require a recent Chrome or Firefox, come up with three common launchers for each platform (think PhoneGap), and then run the Linux backend inside a VM on Windows and OS X. But that seriously restricts the set of apps that are in scope, and at that point, you might also try just building a cross-platform backend (using Go, maybe?).
So most of the question is whether you're happy with this as a distribution mechanism. The best you'll be able to do is something like VMware's Unity mode, which isn't bad, but isn't good either. Alternatively, you could run an X11 server on Windows (via Cygwin) and OS X (Xdarwin), but that's also going to look very non-native.
If you're not, then there's an open technical problem of how to make it look reasonable. I'd consider whether browser-based desktop apps are an option: you can require a recent Chrome or Firefox, come up with three common launchers for each platform (think PhoneGap), and then run the Linux backend inside a VM on Windows and OS X. But that seriously restricts the set of apps that are in scope, and at that point, you might also try just building a cross-platform backend (using Go, maybe?).