It's your money if you want to get a new lens, and it's definitely your own prerogative if you want to do so. But like all everyone above me who have posted (in some form or another); I will like to say it's not only the equipment that matters. I would say your personal shooting style is what matters.
You mentioned that the aperture at the tele end of your 18-200 is a limitation for you. I'm sure you have found out that using f5.6 at 200mm is limiting you in what you want to shoot; but since you did not mention anything much about this; I am making a couple of assumptions here:
1. f5.6 at 200mm doesn't give you enough shutter speed to handhold comfortably?
2. f5.6 doesn't give you a fast enough shutter speed to freeze action?
So to solve that issue, your best choice will be the 70-200mm f2.8. Note the weight when you do your research before splurging 3K++ (IS version) on it though. The 70-300L won't solve this particular issue since its biggest aperture at 200mm is f5.0.
However if you indeed choose the 70-200mm, note that you will lose the wide range of your 18-200mm. Unless you are fine with shooting mainly tele, getting the 70-200 will not get you a general-purpose all-in-one lens.
If you want to replace the range of your 18-200, and since your 60D has a cropped sensor, you might need to get get two lenses instead, one to cover the wider range, if cost is not a factor, i'd suggest to you the excellent 16-35mm f2.8 or the 17-55mm. Then you will have two solid lenses to cover both the wide and tele ends of your current 18-200 range. However, do note you will have to change lenses instead of having a general-purpose all-in-one lens that you can keep mounted on your camera for most occasions.