So what does the diode do?
A basic lithium ion charger is a constant current charger, with an end-of-charge detector, charging ends when the battery reach its full charge at 4.2V (most common) and the charger goes into trickle charge mode.
A solar cell can be modeled as a current source in parallel with a diode. Higher voltage is achieved by stacking these individual cell in series. http://en.wikipedia.org/wiki/Solar_cell#Equivalent_circuit_of_a_solar_cell
In this case, the charging circuit consist of just a blocking Schottky diode. Advantageous is its low forward biased voltage, which is around 0.3V.http://en.wikipedia.org/wiki/Schottky_diode
Is it safe to charge the battery in this manner?
The charging current is very small, comparable to a trickle charger, the solar cell act as a weak constant current source. The panel I brought has a open circuit voltage of 5.8V under bright sunlight and a short circuit current of about 80mA, you can measure this with by simply shorting your digital multimeter across the solar panel terminal, this is slightly more than 10% of battery capacity at 700mAh. But remember when charging occurs, it is at a voltage 4.0V and above (3.7V + 0.3V diode drop), the net charging current (minus the current drawn by the phone) is not going to be any more than the measured short circuit current.
In any case, the rule for charging a battery safely is to do it at 0.1C (i.e 70mA) and here, the charging current does not even excess this figure.
A net positive current in excess of the stand-by power will charge the battery. Some experiment I've done shows promising result. It take 3 hours of good sunshine to charge a 'dead' battery with the phone powered off to last 12 hours when the phone is standby without charging.
Again some calculation, working back, given that the phone draws 6mA on standby, 12h x 6mA = 72mAh, each hour of sunshine gets you 24mAh (72mAh/3), on standby the phone consumes 6mAh, thus a net charge of 18mAh will go to the battery. Consider a day with 6 hours of charging (day) and 18 hour without (night), your phone left on standby will run indefinitely.
It is best however, to let solar panel charge the phone ONLY when battery indicator shows one bar below a full charge.