If I publish an update on Google Play without increasing the version code, will new versions be available for new installations?

I have an application published on Google Play.

My application is not compatible with some screen size (it was not shown to all devices), so I fixed this problem.

Now I would like to publish a fixed application, but without requiring an update for people who have already installed my application.

Is this possible without increasing the version number and updating the fixed APK? Or do I need to do something?

If I publish an update, how long will the new version appear on the market?