Hi, first time user to this website. I hope I can get an answer.

I bought an under the lighting bar for under the kitchen cabinets that has an AC plug. Instead of using the plug I cut off the plug and drilled a hole through the dry wall and ran the wires into the AC outlet. I wired the wire that was from the big side of the AC prong to the white wires in the outlet and the wire that was on the small prong to the red wire in the outlet(there's also a black wire in the outlet and the bridge between black and red is unbroken).
It lights up. But did I do this correctly?

Thanks!

No, cord is not allowed inside walls.

I understand this doesn't meet the electrical standard but in terms of the outlet points which I used is it correct? That is my question. What I can do is run the proper wires from the outlet out through the dry wall and then use join up with the cord outside the wall. How is that? :)

Yes you connected it correctly.

You can go back to the original and plug it in. That is all you can do with the fixture you have.