The detector you show is called an Envelope Detector (Wikipedia).
You are missing a couple of important components: A resistor and a capacitor after the diode.
When the diode conducts it charges up the capacitor which then discharges relatively slowly through the resistor. The time constant is selected to be long relative to the carrier frequency but short relative to the modulating signal.
The resistor and capacitor fill in the signal between the carrier cycles to get back a close approximation to the original sent signal.
With a more complex signal such as audio you might get something like this:
The rapidly changing signal is the carrier while the red outline is the wanted output.
For a simple receiver such as you show you need to be careful that the connection of the antenna and the detector do not disturb the LC resonant circuit so much that it changes the frequency of resonance from where you need it and may also increases the losses to stop it working.
Typically the antenna will be connected to the coil with a small capacitor to reduce such loading and the detector will be taken from a tapping on the coil, not at the end for a similar reason. This will allow the resonant circuit to operate more effectively.
You need what is called a "High Q" in the resonant circuit to help magnify the small signal from the antenna.
Such as in this example:
The diode will actually start conducting at lower than 300mV, probably around 100mV as shown in this chart you get some output evener a few 10's of millivolts of signal.
Even with all this the output signal will be small and you will need sensitive high-impedance headphones - normal low impedance headphones for personal stereos will not work, I have used a crystal earpiece and they work satisfactorily although that was many years ago. You can also feed the signal to an audio amplifier.
There are a number of web sites about crystal radios that can give information to help get yours working - here is one Techlib.com.