I hope you don't mind, I treat problem posts like tech support case notes.

Issue: No crank, no start. Starter clicks, fuel system makes its usual sound.

Before troubleshooting: Battery is brand new
Edit: Specs - 2006 Saturn Vue, 6 cyl, automatic, front wheel drive

Troubleshooting steps taken:
Started on Saturday, tried to get out of a store parking lot. Click, no crank, no start. Everything else lit right up.
Jumped car - no dice
Shifted all gears, tried from neutral - no dice
Replaced starter - no dice
Cut and re-stripped the positive cable (the one going to the lights, electronics, etc.) since it had signs of an overheat in the past - this seemed to work, but it could have been related to how long it was disconnected.
Drove it Sunday just fine.
Monday, didn't work. Click, no crank, no start. Fuel system made its usual sound.
Tried from neutral - no dice.
Loosened and retightened the... "Splice block" I guess I should call it? Where a screw is to disconnect and reconnect a stripped wire, as done Saturday. - no dice
Tried disconnecting positive at the terminal for a while - no dice
Battery terminal connectors seem fine.

What I know I haven't done:
I haven't recut the wire that goes to the starter directly, yet. I think that comes next (I thought I should probably do it to begin with, but my family and I had a discussion about it first)
Since the result is the same with a jump, I don't think the alternator will be a factor here.

What it sounds like to me:
Starter's not getting enough juice? I'm gonna tackle the battery to starter wire when the sun comes up. Then again, it may be time to replace wires. I've seen the battery light come on at random times while driving, even with the new battery. Usually I could turn the stereo off and back on and that would go away, which I find odd.

Sorry that this is so long winded, I like to cover everything I can when I post on a forum.

 

Several things come to mind.

0. What is the security light doing before you crank and while trying to crank?

1. If your charging light comes on and off, it is a sign that the alternator can not put out enough current per unit time to keep the system at a high enough voltage for everything to operate properly. Or the voltage regulator inside the alternator is flaking out and not properly regulating the voltage. I'm going to guess you have an aftermarket audio setup, which, when cranked up, may be a tremendous drain/load on the charging system. That would explain why turning your radio off kills the charging light...you stopped pulling huge currents.

2. It is possible that some or all diodes in the alternator have broken down and are funneling current directly to ground. If this is the case, that battery drain to gnd is constant, including when the engine is off. I would put a multimeter across the battery terminals with car at rest and while trying to crank. It should be about 12V.. My guess is that your battery has been drained below that significantly. If the drain through the alternator to ground is high enough, it could be why the jump didn't work.

You need to get the battery charged back up, via a battery charger or other less safe means. Take it to autozone or similar to have it tested for a bad cell. If bad, obv buy a new one, prorated for warranty ( bring your receipt) If not bad, ask them if they have a charger that you can use at the store.

3 bad cell in battery. Already mentioned.

4 ignition switch issues preventing signal to starter solenoid . But you said you hear the click on two starters, so prob not this.

Measure batt v as described above, post results.
Then go get battery checked then charged. Actually you can bring the alt too for testing.
Let us know what you find

 

Have you tried replacing the ground wire to the battery?