If your talking about current games. Windows 7 x64, the difference between Home Premium and Ultimate is negligible.
Older games, it will depend on how old, Windows 7 will run most things designed to run on Windows, it may struggle with DOS based games. But you can run those in Dosbox anyway.
First you need to figure out what sort of games you want to play.. With Windows its essentially the hardware flexibility which makes the battle heavily in its favor.. You can basically choose what hardware you wish to purchase, even assemble them together yourself, install the OS, install some basic drivers and you are done.. Not so simple with the other OS's.. Also, you can basically keep upgrading your hardware or add new devices without any much hassle.. Again not so convenient with Mac or Linux.. Thus, Windows is the clear winner over here.. As for which version of windows, if you have a somewhat modern PC then Windows 7 64 Bit (anything except starter edition) is the one to get..
currently win7 64 pro or ultimate. very few backward compatibility issues. i only know of 4 games (post 2006) that wont install on it and there not very good 1s. you will have problems with games that install with 16 bit installers but you can VM them with dosbox or similar if you really want to play them.
in a few months windows 8 will arrive and that is supposedly more geared to gaming.