I have a Gateway 5626, and I've recently purchased an Intel Duo core 2 E6850. Their sockets match up and everything, and the BIOs seems up to date. However, when I turn on the computer it refuses to boot up and leaves me with a black screen.
It seems like the motherboard simply doesn't support it. From what I've gathered, your motherboard board doesn't support the E6850's front-side bus link of 1333Mhz, the board supports up to 1066Mhz, so the E6700 is as far as you could go with upgrades without changing the motherboard.
That and I'm not sure what the CPU support is like for the G5626's BIOS, even with a completely physically compatible E6700 it may still not work.