White gold is a mixture of gold, palladium and nickel, generally speaking. If there was any reaction with the gold, you'd see dark colored grayish spots on the jewelery. With yellow gold it's pretty easy to spot, and with the white gold you'd notice the differences in the overall coloring where the mercury contacted it. For the Hg to amalgamate the metal, it really has to be in contact with it for a good deal of time. Just having a drop or two of mercury land on the ring, then quickly wiping it away will not result in an amalgamation.
If you are really concerned about the Hg, just take the jewelery and place it on a high flame in an outdoor grill. It takes quite a bit of heat to melt the jewelery, but only a little bit to make any remaining Hg vaporize away. Again, make sure this is done outside and stand upwind from the vapor. The high temperatures will cause any Hg that is remaining on the surface to evaporate off of the metal.
Now in reality, one mercury thermometer has about .5-1 gram of actual mercury in there. It's not a hell of a lot so if one breaks the 'contamination' isn't something to be very worried about. You don't want to be consistantly breaking thermometers with Hg in them, but having one break on you isn't going to poison you in any real manner. (A good analogy to the exposure of one thermometer's worth of Hg would be like having a little cut on your hand. That little cut on its own really doesn't cause any harm and is more of an annoyance than anything else. Now if you kept adding new cuts to your hand, then you'd have a problem due to the sheer number of them. You would also have a problem if you had some pre-existing condition that made the cut more damaging or unable to heal as quickly).