Current flow due to capacitor in a basic power supply circuit
When the capacitor is supplying current, doesn't it complete a path
  like this.... 





  ....and turn diode "on" during negative half cycle?


No it doesn't. Current cannot flow back thru the transformer secondary winding and then thru the diode because the diode will block this. The voltage on the cathode of the diode is greater than the voltage on the anode therefore, no current flow. A diode is not "turned on" by current, it conducts current only when the anode is more positive than the cathode.

Note - simplifications on diode characteristics have been used in this answer. Leakage currents when diode is reverse biased means a tiny current flows back thru the transformer winding on negative half cycles of the secondary voltage.