I wonder if this is the kind of thing he can make a comeplete recovery from, or whether it will permanently damage his strength to some degree.

I'm trying to think of some position players that had labrum/rotator cuff tears and how they did coming back. The only ones I can find are mentioned in this article. Griffey in 2003, J.J. Hardy in the minors in 2004 and Freddy Sanchez this past year.

Adrian Gonzalez is a recent example. And he's been relatively sapped of power since he had the surgery.

Adam LaRoche in 2011 too. Seems like his power was just fine though.

My first thought was Shawn Green, but Wikipedia says his shoulder injury was tendinitis.

Jim Edmonds had his non-throwing shoulder surgically repaired in 2006, after the season. He then followed it up with his worst season in the majors, but of course he was 37, and IIRC also suffering from concussion symptoms, and he bounced back the next year.

Scott Rolen also tore his non-throwing labrum in the collision with hee-sop choi (2005), and has had at least five surgeries since to try to fix it. He's never been the same, offensively.

Did Shawn Green have a labrum tear and promptly turn into a pumpkin?