I've recently tested to connect the backlight to the 3v pin on the arduino instead.
Regardless of how you power an LED you still need some way to limit the current and this almost always involves the use of a resistor.
Depending upon the LED configuration in the backlight circuit the 3 V supply may not provide enough voltage and/or current to work properly.
You should power the backlight from a 5 Volt supply using a resistor in between the supply and one of the backlight pins. Unfortunately your data sheet does not supply enough information to determine the required resistance so you will have to experiment. I would start out with about 150 ohms and see if the voltage across the LED is then within specifications and if the backlight is sufficiently bright.
Could I have destoyed the backlight when I wired it to the +5V?
You didn't help it or the Arduino power supply at all, but you probably didn't destroy either of them (yet) either.
Don