Maybe, with a few stealthy lawyer tricks
As the recent proceedings between Katie Holmes and Tom Cruise have shown, it is very hard to get your behind-the-scenes actions to stay there. Even in a closed courtroom, the simple fact that you are airing your dirty laundry in front of multiple people decreases your chances of keeping the information to yourself. Also, unless it is a case involving a juvenile, the court record cannot be sealed. However, with a couple of moves, even lawyers not retained by the world's wealthiest movie stars can take quiet action on your behalf.
Of course, every state is different in what they want filed in a judgment. But one of the best ways to keep certain documents out of the court record and out of the prying eyes of the public is to file documents that make reference to other documents that contain key information pertaining to your case. It is a bit of subterfuge but a good way to keep explicit documents out of the limelight.
Another way to go is to just take the case to a private judge or mediator. In either of these cases, the person presiding over the case can make reference to certain documents without explicitly saying what those documents contain. It is a less stealthy route, but also a much more expensive one. However in this case, there needs to be an agreement on both sides that this is the way to go. All it takes is one of the parties involved to say "no," and all bets would be off.