No, the transformer supplies voltage to the diode and then a current will only flow if there is a path for it to flow in.
A diode simply stops the current from flowing in one direction and it doesn't have to go anywhere else. It just stops.
When the voltage reverses, and it is possible for the current to flow again, then it does flow.
It isn't like trying to stop a locomotive. The voltage can be stopped by a small diode if it has enough voltage rating to do it.
In fact, because it doesn't conduct any current (other than microscopic currents) in reverse , the diode does not even dissipate any power, so it doesn't get warm while it is stopping a high voltage.
A diode costing just a few cents can stop a voltage of up to 1000 volts.