A voltage divider is inefficient, but it also will not provide a stable output voltage. This is because the current drawn by the PIC (which includes the current drawn by the LED, which flows through the PIC) affects the currents in the voltage divider, and this affects the output voltage.
In this case, the LED is the main load on the voltage divider. If you adjusted the voltage divider resistors to give roughly 5.5V when the LED was ON, then when the LED turns OFF (or tries to turn OFF), that load current will disappear, and the voltage divider's output voltage will increase.
A voltage divider will only produce a correct and stable output voltage if the current being drawn from it is a lot lower than the current flowing in the voltage divider resistors themselves. You can increase the "stiffness" of the voltage divider by reducing the resistor values, but this wastes more power in the voltage divider.
The simple answer is that voltage dividers are not normally used to provide power supply rails to circuits; an active regulator is needed. Using a 78L05 will reduce the wasted power, and increase the regulation of the PIC's power supply voltage.
Yes, it's possible that you have damaged the PIC with overvoltage. This is especially likely if you used low resistances in the voltage divider. What values did you use?