A Tiny Text File Read/Write class Implementation in C#
Naming

FileIO: The File class resides inside the System.IO namespace. FileIO is quite a misfortune chose of name. TextFileWrapper, TextFileUtil or TextFileManager would be a more appropriate name for this in my opinion.
FileWrite: The general guidance is to start your methods with a verb. So, WriteFile would be a better name for this.
ReadTxtFile: This name is a bit misleading. txt can be considered as a file extension, so the caller may no specify the filename as you expect.
inputString: Please try to avoid Hungarian notation.

Please try to chase consistency (or symmetry).
Try to name your Read-Write method pair in the same fashion:

ReadTextFile, WriteTextFile
ReadFile, WriteFile
or simply just Read and Write

Stuttering:

Maybe it's just for me, but FileIO.Instance.FileWrite seems too repetitive.
TextFileManager.Write seems way more natural for me.

Robustness
Your implementation is quite naive. It does not deal with a lot of different cases:

What if the fileName points to a non-existing file?
What if the inputString is enormously large?
What if one of the parameters is null?
What if the file has been opened by another method and has not been closed yet?

Your implementation also relies on Encoding.Default, which is computer specific.
Which means you can end up in the following situation:

FileA.txt has been written to the disk on MachineA
FileA.txt has been copied from MachineA to MachineB

where the default encoding is different


MachineB could not read FileA.txt

Thread-safety

Your instance property does not make your class a singleton.

Please read thoroughly Jon Skeet's guidance.


If you make your class a singleton you have to make sure that your operations are thread-safe.

Reading and writing the same file from different threads are handled in a proper way.


As pointed out by others you have to do some clean-up as well.

To make sure that if something goes wrong then your file handles are closed properly.



Performance

Try to take advantage of async I/O
Try to take advantage of ArrayPool when you are calling the GetBytes method.

It has an overload which accepts a byte array


As always, measure, measure and measure.