Parts of it may collapse inwards a bit, if they were only held together very tenuously. Overall the effect would be rather pleasant, smoothing a surface that otherwise is likely extremely sharp and irregular.
In fact, you may way to have people go around and pound down the surface a bit, as walking on it would probably be akin to walking in a field covered in large glass shards. Dangerous for the feet, and even more dangerous if you need to keep your space suit contained.
Note that if you choose to give the asteroid some type of warm atmosphere as well, that would have a huge (and possibly destabilizing) effect on the asteroid.
Thinking about this a bit more, I think you might WANT an asteroid that has a noticeable amount of frozen water (and/or other liquids/gasses that solidify at deep space temperatures) in it. If we assume that your gravity generator is in a sufficiently shielded capsule in the middle, once you add in your atmosphere (however this occurs) the asteroid would be ripped violently into pieces - and that would be desirable. Specifically that process could be used to accelerate the decomposition of the sharp rocky parts of the asteroid, which would probably have quite a time escaping a body with the same gravity as earth. Sure it would fire off like a rocket, but it takes a REALLY powerful rocket to escape Earth's atmosphere and gravity. After a waiting period gravity would settle things out and I would expect that, on average, you would have fewer large sharp pieces.
You would probably end up with some significant sinkholes-in-waiting, but that's likely to occur naturally anyway. To counter-balance that, if you were to increase the gravity of the asteroid SIGNIFICANTLY during the process you would both decrease your wait period between explosion and settling AND decrease the number of unstable/thin-shelled holes in the resulting body.