Can mercury react with water? If so, what is the reaction?

Mercury does not react with water. Mercury and water do not mix, as mercury is almost 5.43 times more dense than water, causing it to sink to the bottom of a container if the two are mixed. 
Mercury can react with oxygen molecules at high temperatures, so if the water were to be chemically altered, such as through electrolysis, the mercury could bond and form a salt. Mercury can also react with some acids, so if the water is impure, there may be smaller reactions occurring and corrupting your mixture.
Because mercury has such a high surface tension, if water and mercury are mixed and agitated, you might be able to cause small beads of mercury to separate themselves from the main body of mercury, but this is a physical change, not a reaction.