In the basement I am trying to add two canister lights. I have installed the lights and added the whips. These are 120v lights and the whips have three wires: black, white, green. First thing I did because I thought it was going to be so easy and I am not an electrician is open the nearest junction box and wire it up. I didn't put everything back together yet. I turned everything back on and found I had wired into an outlet. UGH.
I have a 600v wire running through the basement ceiling that connects 6 fluorescent light fixtures. This wire has 4 wires: red, black, white and green inside the cable. Can I splice into this in any way?

First you don't have a 600v wire running through your basement. You have a wire with insulation which is rated up to 600v. Second you will have to tell us more. Like how the wiring is run. If you have four wires it sounds like you have a switched and an unswitched leg in that cable. Whether you can tie in at that junction box depends on whether the switch comes before or after the junction box.

One other thing of great importance.

Where is the junction box thatyou wired into. According to the NEC. The junction box has to be readily accessible, (not behind walls or ceilings).

It sounds as if you were surprised to find out you created your splice to a wall receptacle, when in fact you should have placed your new wire directly into the receptacle's device box.

Can you please attach some photos of what you have and what you are trying to do?