I didn't see the ring in the second photo...but closer view of the pic shows a distortion in the area (also on the pad of the thumb.)
If the first photo can be linked from a news site, I'm a complete believer!
If it's the same ring...
Chandra may have given it to him. He may be wearing it in some kind of dopey attachment to her.
He may have taken it off her body. Why? Thinking that it would deter identification of her (not well thought out plan, but such happens a lot in the panic of crime). He may have taken it off to make it look like a robbery, along with the watch, but in dopey sentimentality kept the ring. Remember, he thinks he's smarter than everyone else. Might also be a Sherlock Holmesian psychology, that criminals unconsciously want to be caught, so they give clues.
As for the watch, he may hide the box to stave off a connection to the watch. But if the watch was not found, and he disposed of the watch elsewhere, why would the box be such a big deal? On the other hand, if the type of watch was publicized before, and with the threat of a search, he recognized there was a piece of evidence in his home connecting him to a "Tag Heuer". His incompetent disposal of the box may be sherlock holmsian too.
Or, maybe, creepy/guilty son of a preacher man he is, he is just covering up his affairs in his blundering way.
Many permutations and possibilities. Ping me on ring stories.
Interestingly, the Modesto Bee story today about the ring pairs a photo of the design of the ring with a pic of Gary - and of all the pics possible, they use the second Karate Chop one. Is the editor saying something to us???