I know this thread is old, but it may help someone else out there.
I would recommend changing the battery. My battery was completely dead during a sub-zero cold snap...nothing but clicking relays when I tried to turn it over. I jump started it and it fired right up, but would stall the second I took my foot off the gas. It wouldn't stay running at idle. The problem was immediately fixed by installing a new battery, and there hasn't been a problem since.
Here's my question....why would a bad battery cause a stall, after the engine is running and the alternator is providing power? According to the mechanic at my local Jeep dealership, it could have something to do with the PCM (Powertrain Control Module). He said the PCM loses it's "memory" when the battery is disconnected, so the car may need to run for awhile in order to give the module time to relearn the proper idle settings and reset itself. But the stalling issue went away immediately after installing the new battery. It ran fine at idle right from the get go.
I think something else is going on, but I don't know what. How does a dead battery affect the electrical system when the car is running? How does it affect the PCM? It anyone has an ideas, I sure like to hear them.