Have you had an issue where desktop Java applications broken inbetween releases of the same Java version? (ie, from Java 7 u60 to Java u72 or something). Otherwise its an unfounded myth.
The single biggest problem you're going to have with Java is version-of-the-week hell. Write once run anywhere is a joke.
I've run my projects in more VM versions of OpenJDK and OracleJDK that I can count, played Minecraft with all the damn updates that came every two weeks or so, never had an actual VM compatibility problem. They do take retro compatibility very seriously (to a fault even), thus why no API was ever actually removed from the runtime.
Hell I've never in years had an issue where Eclipse would crash because some VM was incompatible, and Eclipse its a massive application.
And please, don't even mention applets. They shouldn't exist, its a Good Thing™ most of them stopped to work.
In any case, there wouldn't be any single damn problem if you just provided a link to download the JRE but Oracle is composed top to bottom from a pile of steaming stinky assholes and they bundle crapware with their damn JRE distributions (luckily they dont do it with the JDK). So yeah, bundle a JRE (20Mb to 40Mb, libGDX guys provide a tool to reduce the size of the VM by removing unwanted crap).
Still this is an issue that you will have in some measure whatever you choose. C# needs the .NET runtime (or Mono depending on the platform, which is a whole other issue altogether), Java needs the JRE, C++ will need whatever MSVC runtime you're using (or some specific glibc version depending on the platform), etc.
The answer to all of those is: Ship the dependencies with your application, and save yourself a headache.