It's difficult to prove a negative, but I believe there is no evidence that MPs on landing craft shot deserters or soldiers that did not disembark. No accounts that I've come across include it. Not the US Army official history, not modern accounts like Rick Atkinson's Guns at Last Light. The general consensus is only one person in WWII, Eddie Slovik, was officially executed for desertion by the US Army, and there isn't any evidence for widespread unofficial executions either.

MPs attached to assault units were primarily there to direct traffic and control POWs taken, the same purpose as they would serve during the rest of the war. The US army maintains a brief description of their role at Omaha beach here