LEDs are driven with a constant current source. Adding a relay with a constant current source is one way to power an LED, and probably not the cheapest or simplest way. However, you mentioned using a relay, so I'll assume that you need to use a relay for isolation or something.
A relay needs fairly significant current levels to energize the coil. It is usually powered with a constant voltage source, unlike an LED. Furthermore, this voltage must be in one direction, that is, it must be a DC signal. On small relays of the type which might be used to drive a typical LED, this creates a current which is typically about 50-100 mA. An audio signal doesn't have enough power to drive this kind of current or voltage.
Additionally, audio isn't a DC signal, it's an AC signal. The cell phone audio port is likely to have an output capacitor to be incapable of outputting a DC signal. To detect the presence of audio, then, you'll need to rectify the audio signal, that is, to convert it from AC to DC.
At 0.6V output levels, a simple diode rectifier won't work, those need significantly more headroom. You may be able to get away with a Schottky diode rectifier if costs are tight, but this will need to be amplified. Again, you may be able to get away with DC-offsetting the output of this regulator to slightly less than the threshold voltage of a FET or the base-emitter voltage of a BJT, and slightly more than the required voltage when increased by 0.2V, but that would require some precision where there's usually not much precision.
If cost isn't a huge factor, it will be far easier to get yourself an op-amp and build a precision rectifier like the following example:
As described on the Wikipedia page, the output of this system has a gain of \$ R_2 / R_1 \$ when the input is negative. This output can then be used to switch a transistor, or, with a sufficiently strong opamp, to drive the relay directly.
SSRs use triacs as switching elements, and those are for AC, not DC. A triac is like two transistors so arranged that, once ignited, they keep each other switched on. The only way to switch it off is to reduce the current to below the hold current. AC devices do this 100 or 120 times per second, at zero crossings of the mains, but at DC it will not switch off.
Instead of interrupting the circuit you can also briefly short-circuit the triac's terminals to switch it off.
Best Answer
From the diagram on the SSR (as given in the datasheet), it uses a phototriac. The SSR will not deactivate until the load current drops below the sustain current for the device, which is not given in the datasheet but is usually very close to 0 regardless. This device is not suitable for controlling constant DC loads.