I wouldn't go lower than the 7850 for 1080p and that is pushing it. The 6850 is getting creaky.
The 6850 is fine for 1080p. Probably the lowest I'd go for that resolution, but still fine. Obviously this guy is on a budget, so adding on another $100 for a much, much better card is probably not an option for him. That being said, of course get the 7850 if you can afford it.