I do not know what the U.S. is doing behind the scenes to try to resolve the issue. However, if Mr. Davis has diplomatic immunity (there is some uncertainty concerning his status), the U.S. could take the matter to the International Court of Justice, if Pakistan fails to adhere to its obligations under the Vienna Convention. Outside of that, the U.S. could seek to apply pressure e.g., reducing/eliminating aid, but there would also be adverse consequences to that move i.e., Pakistan could terminate the ability of the U.S. to use Pakistan's territory to provide supplies to its forces in Afghanistan, could suspend intelligence cooperation, etc. Some in the U.S. government appear to be making threats along the line of reduced or suspended aid, etc.
My guess is that should the U.S. guarantee that the shooting would be examined within the U.S. legal system with Pakistan's being permitted to supply evidence/argue its case/have full access to the proceedings, that could help resolve the dispute without a resort to the ICJ, much less policy pressure that could also damage U.S. interests. Nevertheless, if he has diplomatic immunity, Pakistan is obligated to refrain from prosecuting him, though it can expel him.
If, however, Mr. Davis actually lacks diplomatic immunity, Pakistan is within its rights to prosecute him there and in accordance with its own laws. Under such circumstances, it would also be within its rights to apply its own criminal penalties if he is convicted.
