As of September 2016, No.
All current implementations of Docker on Windows use virtualisation,
Docker 1.12 uses a hypervisor in Windows -thus removing the advantage
of containerisation over virtualisation.
Docker needs more than just using Linux systems calls.
It needs process control groups (cgroups), a stackable filesystem
(aufs), plus other Linux-based systems outside the kernel.
Neither cgroups nor aufs are natively in the Windows 10 kernel.
There is an implementation of Windows Server 2016 here:
https://msdn.microsoft.com/en-gb/virtualization/windowscontainers/quick_start/quick_start_windows_server
..but this will only run some Windows services, e.g. IIS, and not
Ubuntu
The source (thanks to muru) has answers with more detail and insight.
Thomas stated it quite well and short:
I'm fairly certain WLS doesn't have all the necessary infrastructure to set up Docker. The core issue is that WLS, while having Linux-isms, is not really truly full-featured Linux. This is something that many are just accepting as a limitation…