When you say that it doesn't turn on, is the notebook not turning on at all (no fan noise or power lights) or is it just not displaying a picture on the LCD screen?
If the computer turns on but does not display a picture on the screen, press and hold the
fn key and while holding the
fn key down press and release the
F4 key several times to see if the picture is restored. You may simply have the computer set to display on an external screen only. If so, this will fix it. If you have access to a standard desktop monitor (CRT or LCD) you can try connecting it to the external VGA port on the back of the notebook to see if the picture displays properly there.
If the FN-F4 combination does not work, try shining a small flashlight at an angle across the screen. If you can see a faint image of the normal Windows display on the screen, then you may very well have a failed backlight in the screen that will need to be replaced. Replacing backlight bulbs in normal LCD screens is a fairly painless process, but touch screens are, if you'll forgive the pun, touchy

and I wouldn't try it unless you know what you are doing. If the computer is still under warranty, give Gateway a call and get it fixed. If not, look for a local certified tech to replace the bulb.
If the computer does not turn on at all (no power light, no fan spinning), try what Robisbell suggested and remove the battery (disconnect the AC adapter before you take the battery out), then connect just the AC adapter with no battery and see if the system will turn on. If it does turn on with the battery removed then you probably have a bad battery. If it does not, then you likely have a failed AC adapter or internal power supply. Once again, if the system is under warranty give Gateway a call. If not, I would recommend having a local repair tech take a look at the machine to verify that the AC adapter and internal power supply are functioning before spending money on a new battery.