Amount of memory is a minor consideration compared to the processing power of the video card. A higher resolution = larger amount of pixels = more processing power needed to render the games.
The 9600GT is not intended for high resolutions no matter how much memory it has. It will do fairly well at 1280x1024 and below but above that you will need to turn down the settings a lot in most current games.
If you are talking about 1920x1080 then the minimum I would recommend is an HD5770; http://www.newegg.com/Product/Product.aspx?Item=N82E168...
There is no hard and fast rule, it depends on the games and what settings you are comfortable with.
Resolution itself has little to do with memory requirements. It is once you start dealing with large texture files, anti aliasing and and other effects which require data to be switched between the memory.
The 9600GT can be used to play games at 1920 x 1200 using medium quality at best.
I actually played Crysis using my 9600GT on my 26" monitor. It was playable and speaking from memory, it seemed like I was getting about 28 - 30 FPS on average, except very demanding parts of the game like the final battle scene. It was kind of a fast slide show at that point, but even higher end cards struggle with the final battle scene.
I tweaked my setting to be a bit over medium quality however:
1. Since I am still using Win XP, I was playing it in DX9 mode.
2. My playing style is not close quarters combat. I generally pick off my enemies from a distance and I use a decent amount of stealth. Of course, sometime there is no other option but close quarters combat.
Generally speaking, if you're resolution is above 1680 x 1050, then you should have 1GB of video RAM. However, the video chip itself has to be powerful enough to be able to use that extra VRAM to take advantage of it.
For example, the HD 5670 1GB card is barely faster than the HD 5670 512MB card. In fact you need benchmarks to see that it actually does "perform better" than the HD 5670 512MB.