"Energy harvesting" from RF and other very low power sources (like motion and light) is becoming a big field in electronics and there are lots of parts coming onto the market that make this easier.
Here is an article specifically on harvesting power from ambient radio waves...
The amount of harvestable radio energy is widely variable and depends on factors such as how strong the transmitter is, how far away the transmitter is, what the frequency of the waves are, and what kind of antennas you the receiver and transmitter have. When I was a kid, a ham radio operator showed me a trick where he lit up a florescent light bulb from across the room using the power from his transmitter, which means that several watts worth of power made it to the bulb. On the flip side, modern radio receivers can be sensitive down to nanawatts (1/1000000000 or a watt). A very wide power range.
LCDs would be a good choice for an application because it is possible to drive them with very little power. That is why they are used in watches which run for years on a tiny battery. It is not uncommon to find small LCDs that can maintain an image with less than 10 microwatts of power. A microwatt is 1/1000000 of a watt!
Considering the tiny amounts of power you are likely to have available, you would probably have to make trade offs between the functionality of the device and how much power it took to support that functionality. One common way to reduce power requirements is to reduce the time the device stays on, so something that it only on for 1 second per day has the other 86399 seconds each day to collect the power it needs for the brief moment when it is awake. Of course, a device that is only on for 1 second per day might not be very useful.
Long story short- it is theoretically possible to create device that is powered only by radio waves, but it would take a lot of careful design and engineering to make one that is useful.