All I had handy to compare was a 6950 to a 7970, but they look like the same distancing on the X-Bracket. I'd say if the heatsink only makes contact to the GPU die, and sits over all the capacitors without bumping then go for it. My only concern is if you don't end up with any direct cooling for the Memory or VRM.
If the 7850 had no prior cooling on those areas, and the cooler from the 6850 fits without conflict, go for it!