Mostly because:
The Royal Navy had naval supremacy over Germany. While Germany had partial air cover over the Mediterranean from bases in Italy and the Balkans, it was simply too risky to try a large scale naval landing with the Royal Navy in the way.
Germany didn't have the transport capacity to land enough troops to make any real difference. For example the Normandy landings used 1,213 warships, 4,126 landing craft of various types, 736 ancillary craft, and 864 merchant vessels to land only 150k troops on the first day. This was larger than the entire Kriegsmarine, and 150k troops is about a single battle's casualties on the Eastern Front.
Germany could never have kept them supplied, for the same reason as #2. Any troops landed there would have been pretty much immediately cut off and annihilated.
It's debatable whether or not Turkey would have allowed German ships to pass through the Bosphorous Strait
Germany had neither the Naval dominance, nor the logistical capabilities to launch such an undertaking. The conquering of Crete for example involved little to no maritime operations by the Kriegsmarine or Italian Navy. Rather they utilized large quantities of Fallschirmjagers to try to take the Island from the Allies. The British maintained control of both Gibraltar and Malta. Malta is off the coast of Sicily! The Germans never made a high commitment to the Kriegsmarine. Their ability to project naval force was not expanded beyond the UBoats they loved so much.
But really, the major factor is the logistical demands.