Holy crabcakes -- I step away for a few hours and all heck breaks loose 
You are being pummeled with so much misinformation, I'm not surprised you're confused.
Pull up/down resistor are not needed if the pin is set to OUTPUT with pinMode.
An Arduino pin has an internal mosfet switch to VCC and a mosfet switch to ground.
Maybe your Arduino pin is damaged (assuming you did set the pin to OUTPUT).
This is absolutely correct. So, it's not clear why adding a pull down resistor would make a difference. The TC4427 datasheet gives a worst case max input low voltage of 0.8V. In other words, as long as the voltage on TC4427's pin 2 is 0.8V, or lower, in relation to TC4427's pin 3, the output should go low. But, there's a thought -- have you measured the voltage on pin 3 (right on pin 3), with the negative probe on the Arduino ground pin [in other words, the positive probe on pin 3 and the negative probe on at the point where Gnd is supplied by the Arduino? If it isn't 0, then we could be on to something, there. Could be some sort of ground loop [doesn't seem likely with the relatively small current caused by the LED -- but who knows?] .
The voltage relative to the gate would be -12V.
I still think you should try connecting the "+" side of a 9V battery to the GND and "-" side to the gate
Totally ridiculous! The IRF3205 is an N-Channel MOSFET. And why are we talking about "voltage relative to the gate"?!? The datasheet specs voltages relevant to the gate, as relative to the source. And these are positive voltages. The IRF3205 datasheet specs the RDS(on) [of 8mΩ] at 10V [POSITIVE 10V] -- which, BTW, is a good indication that this is truly not a Logic Level MOSFET BUT, esgeroth has stated that he/she intends to drive it with 12V on the TC4427 VDD pin, so that is quite sufficient to drive this MOSFET.
BUT, to get a fix on WHY this thing isn't working, esgeroth is running it at 5V, which, according to the Datasheet, should work.
BUT, it would be informative for esgeroth to apply 12V to the TC4427 VDD pin, and then change R1 to something more like 420Ω.
I looked at an old datasheet, and while it has min/max input values relative to VDD, it only lists a max VDD of 21VDC. All the characteristics are listed using a VDD of 18VDC.
Then that datasheet is bogus. The datasheet esgeroth provided shows this MOSFET to be a 55V device, with voltages like 55 and 44 used in the characterizations.
Have you got your test circuit like this, and running the blink code with 3000 delay?
*That's a good idea. At such a low frequency the Arduino output will be more in line with your manuel test, where you physically connected 5V to the input, then 0V to the input. *