Vulkan API means games on Linux

Solution
In my opinion, no, because developers are going to always focus on the OS with the highest market share first, which means Windows/DirectX. Developers aren't going to not maximize Windows performance just so they can more easily release a title on an OS with less then a 1% market share.

There's a reason why most Linux ports are pushed to third parties; devs really don't want to have to deal with Linux technical support, so it's easier to just push the work (and responsibility) onto a third party.
In my opinion, no, because developers are going to always focus on the OS with the highest market share first, which means Windows/DirectX. Developers aren't going to not maximize Windows performance just so they can more easily release a title on an OS with less then a 1% market share.

There's a reason why most Linux ports are pushed to third parties; devs really don't want to have to deal with Linux technical support, so it's easier to just push the work (and responsibility) onto a third party.
 
Solution
In my opinion yes. Vulkan isn't designed specifically for an OS but works on all of them if you have hardware that is vulkan complaint. It is therefore possible to have one set of code that works in many environments. This allows developers to design once, and then resell to every platform that has hardware that can support it. It means that developers don't need to be experts developing for Nnvidia or AMD/ATI. So as a publisher looking to make profit from selling games, you can sell it to any platform without spending a fortune on cross platform development.
 
With Vulkan its a bit easier to port or cross develop Games for Windows, MacOS and Linux based systems (including Android). Of course all other problems won't be solved with this, but at least one big problem is eliminated, where developers can focus on Vulkan, instead of DirectX and OpenGL or Metal.

Currently, even with that low percentage of Linux gamers, some AAA are coming to Linux. Here is a list of all Steam games for SteamOS/Linux:
> http://store.steampowered.com/search/?term=&sort_by=_ASC&os=linux&page=1

 
The only thing standing in Vulkan's way is developer support for it really, and since it pretty much supports all OSes, and is even supported in both AMD and Nvidia drivers since 2/16/16, there's no reason for them not to anymore (at least the large teams anyway).

Doom 2016 showed very good results with Vulkan. Yeah the game has an occasional freezing issue, but smart devs will know that's not Vulkan's fault, because it happens whether using OpenGl or Vulkan. I've also read recently that it's really only an issue on older model AMD cards too.

The way I see this playing out, is a lot of big budget AAA games will soon support Vulkan. That is because Vulkan requires a MUCH bigger learning curve than other APIs, so it takes an experienced team with lots of programmers to make using it worthwhile.

Some are feeling that also means Vulkan will give Linux/SteamOS a chance to catch up to Windows game library, but there's nothing to support that. If anything, Valve's already had a hard enough time getting Linux converted games to run as well as their Windows versions (usually they don't), so SteamOS/Steam Box interest has waned in the process.

Plus MS are really talking more in terms of console/PC compatibility, including Scorpio spec that will render it as powerful as a 980 Ti, with KB/M support. It would have been hard to imagine at the launch of original Xone, that MS would be better poised today than Valve to support forward thinking PC gaming, but they really are.

Bottom line, the future looks bright for Vulkan, and even might change Nvidia's GPU architecture once it catches hold, which is very likely given that AMD currently holds a lock on console APUs. But Linux's future, not so much, and largely due to Valve not doing well with SteamOS games.

Now granted, I know it's harder to get Windows games converted to Linux to perform well, than it is when giving them Linux support at launch, but I'm really talking about the fact that most gamer's have been negatively influenced by poor performing SteamOS games.

Thus I feel most devs will consider it too much a risk to put time and money into Linux support, because most gamers won't trust it to play well. It doesn't matter if SteamOS is free, it only matters whether devs feel it's worth their dev time, and I don't think most will.

Forget about Windows games converted to SteamOS though. Clearly Valve is fighting a losing battle on that, and that more than anything is what's all but rendered SteamOS and Steam Box meaningless in the gaming world. Then there's trying to keep up on Linux driver support, which is a battle in itself.