Luckily we don't have the Fuel Pump Relay issue that the Grand Prix has (Had a 1997 GTP...heavily modified...and this relay issue is VERY common to them) but you can test it fairly easily to remove that remote issue as a possible culprit to your problem...
This is our under hood fuse box and what each fuse/relay does. Relay 19 is your fuel pump relay and 18 is for your trunk release. Now, I'm not 100% certain but I believe these are interchangeable. You'd need to read what is written on the top of each relay to verify. (Don't have my car to double check this.)
Saturn Sky (2006 - 2009) fuse box diagram | Auto Genius
Here is a picture of the under hood fuse box. Note in this picture, the box is turned 90* clockwise from the diagram in the link above.
IF the relays are the same, you can remove the fuel pump relay and replace it with the trunk release relay. If the car starts like normal, then it's the relay. If it doesn't, then you know your problem lies elsewhere.
You'll see from that picture above there are many relays that are identical to your fuel pump relay that you could swap it with, just be sure to take a good picture of your fuse box before you start swapping out relays so you can make sure you put all the relays back in the correct orientation as they were in when you took them out. You'll note some relays...even though they are identical numbers wise, are in their sockets 180* when compared to the Fuel Pump relay. (The glare is so bad on the trunk release relay in that picture I can't read it's numbers. The Fuel Pump relay has the same numbers as the four relays at the top left of the image...which are the fog light, horn, low beam, and high beam relays.)
The issue with the GP relay was that the fuel pump could be ran at two speeds, low and high. When The GP's relay went out, it was because the low speed setting burned out. The low speed would be used at idle and cruise. The high speed would be used to prime at key on and under Open Loop conditions (heavy load conditions or WOT throttle). So the GP's system would prime just fine, let you start the car then switch over to the low speed setting and the car would die. Even revving it a little usually didn't keep it running.
The fact that you can keep it running by giving it a little throttle leads me to believe it's not the relay or the fuel pump. I don't know if our relay is just on/off or if it has different speed setting like the GP (@Hoosier GXP or
@TomatoSoup might know) but I don't think giving it throttle would keep it running even if it did. It's not the pump because if it was it would never start and never run no matter what you did.
Like I said though, to eliminate the relay being the issue is a fairly easy and free thing to do so I'd give it a try just to eliminate the possibility.