No, you can't. (Well, it might be possible, but very unlikely, that it would work.) Recovery discs are made for specific machines. There are a couple of reasons why the discs for one system can't be used for another brand or model.
In addition to installing the operating system (Windows), recovery discs also include drivers for the machine's included hardware (video, sound, media card slots, etc.). The drivers on the recovery disc for an eMachines system won't have what a Gateway system needs, unless they were built using exactly the same motherboard and other parts, which would be quite unusual. If wrong hardware drivers are installed, Windows may not start and give you the famous "blue screen of death". Or Windows will start, then begin finding new hardware and asking you for drivers you won't have (although you could find them at the Gateway website). The discs also include the applications the manufacturer preloaded on the system. Some of these may not work and cause problems because of the hardware differences (a sound control program with the wrong sound hardware, for instance).
The other reason is that a restore disc also includes information that gets placed into Windows so the system comes up already activated and ready to go. Activation is based on information about the hardware configuration of the system, and yours won't be correct. Windows will see the this and you'll need to activate manually. If the Windows version on the eMachines recovery disc isn't exactly the same as the one on the Gateway, the product key on the computer won't work when you try activation.
Rather than try using the wrong recovery disc, see if you can borrow or download the installation disc for the Windows version specified on the product key sticker on the Gateway. Then you can do a clean new Windows installation and activate it using the product key. You can find the drivers your computer needs at Gateway's website as I mentioned before. It's more work than just running the recovery disc, but it will work. This does mean the system won't have any of the preloaded applications originally on the computer when you're finished. But this is not a bad thing.
Much of the preloaded stuff on a new computer is often called "bloatware", which you really don't need. It just slows the computer down, uses up Windows resources, and may be gathering information and sending it to sites behind your back. You can find good free programs to do anything originally installed and install only what you really need and want. Here are a few suggestions to get you started:
CD/DVD burning: CDBurnerXP
Audio processing: Audacity
Image processing: GIMP
Office suite: OpenOffice
Good luck getting the system back up and running. Thanks for using Fixya!