As someone who has recently released a libGDX game on Steam, unlike .NET and DirectX, Steam does not provide an option with your application to automatically check and download the version you need of those libraries.
It's obtrusive to the user to make them install Java as a system library, as most still view the JRE, especially Oracle's, as some kind of adware/security hole and do not want to have it installed on their machine. There is still a stigma towards Java applications on the desktop, so you have to be considerate of the user.
What I recommend is using libgdx's packr application, which will automatically pack your jar and resources with an appropriate, private OpenJDK which is open to use and distribute with your software. Alternatively, you can do it manually by grabbing the unofficial OpenJDK build that they are using in packr and including it with your project.
I, personally, do that latter option as I have downloading the latest version and packing everything up, including distributing to steam, all included in my project's gradle build script.
Including your own JRE/JDK is lightweight, only adding about 80mb to your game, and is non-invasive. This also insures that the version of Java that you are running your application with is always the version that you need, instead of conflicting with what the system has.
Edit: March 13, 2017
For a little over a year now I've been using Azul's OpenJRE instead of the unofficial builds from alexkasko. They're widely used and supported even in enterprise environments. Additionally, they support Java 8 and up, where as at the time of my switch the unofficial build repository versions were stuck at Java 7.
Since I switched, there is now an open-source repository as well, that continues to provide community built versions of the openjdk/jre, if you wish to use that instead. Both are good.