Will this 'radio receiver' work?
The idea is correct, but it will not work in practice.

If you had an ideal diode (conducts current in one direction with no loss and no voltage drop) and a perfect speaker then you would probably pick up some noise.

A real diode, however, requires a certain minimum voltage to work.  This can be difficult to reach with small antennas.

The required voltage is lower if the current through it and the speaker can be kept very low.  This requires a special earphone that operates on very little current.  You can buy magnetic earphones with very high impedance coils.  There are also special earphones with a piezoelectric crystal inside that will work properly.

If all you want is to receive the ambient "static" radio noise, then two very long wires (tens of meters,) a germanium diode, and a piezoelectric earphone should work.

To actually receive an amplitude modulated radio broadcast, you would need to use a filter made of an inductor and a capacitor (either or both of which may be tunable) to select the radio frequency you want to receive.  

At that point, it is probably better to buy a crystal radio kit.  It will include all the parts you need to build an AM radio receiver.  If you tune it in between stations, you would hear the expected "static."

The problems with your attempted "receiver" circuit are as follows:


The antenna wires are too short to receive enough power.
The diode requires a higher voltage than you can achieve with your antennas and the speaker.
The speaker requires too much current to operate, which drags down the available voltage.  At the same time, the current that could flow would cause the diode to require a higher voltage to be able to conduct.




Under the correct conditions, your receiver would work.

I worked with a fellow who grew up in West Berlin during the Cold War.  There were several extremely high power amplitude modulated short wave radio stations in West Berlin at that time.  Depending on who you ask, they were there to provide news from the rest of the world to those living behind the "iron curtain" or to broadcast capitalist propaganda to dishearten the loyal comrades of the soviet state.  Whatever.

At any rate, this person who grew up in West Berlin told of "building" his first AM radio receiver.  It was much like yours, but simpler.  Just hold one pin of a diode in your hand.  Touch the other pin to one terminal of a speaker, and touch the other terminal of the speaker with your other hand.

In many parts of West Berlin, that was all you needed to listen to those powerful broadcasts.

If you take your circuit to someplace close (hundreds of meters or less) to a high power (tens of thousands of watts of transmitted power) broadcast station then you would probably hear something in the speaker.