2002 saturn sl2 stalls if I take foot off gas after driving for a few minutes then stopping - or after turning car off for a few minutes, it starts right up but I have to keep my foot on the gas pedal. I notice the temp gauge showing the engine is getting hotter - but after about 20 min of sitting there idling (in park) the needle drops to where it normally should be when the car is warmed up and I can remove my foot from the gas pedal and it idles and drives fine after that.

SOURCE: car hard to start after engine is warm

Sounds exactly like what I was dealing with a while ago. A nuisance (especially at fuel stations & such, where if you don't get it right away you end up looking like an idiot as it just turns over & over).

For me, it was my fuel filter. The previous owner had neglected the car so much & I had no idea how long it had been. $50 at an Oil Can Henrys & it was all back to normal. Great first step.

SOURCE: 2002 Saturn SL1 rough, sputtering idle - cylinder #1 misfiring

spray some carb cleaner on the passeger side of the intake manifold, right behind the alternator. it is very common for those to leak there and cause those exact problems

SOURCE: 2001 Saturn Sl2 wont run after warmed up

SOUNDS LIKE YOUR CATALYTIC CONVERTER IS DYING

SOURCE: 2002 SATURN SC1 IDLE JUST STARTED BEING TOO LOW

change the TPS also check the air idle control valve

SOURCE: 98 Saturn SL2 - Stalling issues

I had a problem with starts on my 98 sc2, did it twice only, but it would start up, shake a bit then shut down like it wasn't getting fuel, or a wrong mixture. Replaced the coolant temperature sensor and starts / idles like new now... goodluck