Hi copiman,
I'm afraid you are probably right. On the HP G60 it's most probably the GPU thats failed (or sometimes, the solder on the motherboard for the GPU).
If you have two modules of RAM, chances are that both haven't failed at the same time. So take one module out (remember to remove the battery first), and try to boot the machine up with the one module of RAM, then try putting it in the other RAM slot if that doesn't work. Then take that module of RAM out and put the second one in, and repeat.
If you're getting nothing on an external monitor then the problem is not with the laptop screen, and if swapping the RAM around doesn't help, then it's not the RAM. That leaves motherboard, CPU and GPU, in which case, you're probably better off getting a new laptop. Some people get their GPU resoldered onto the board (reflowing), but this is not worth doing in my opinion, for one, ofte the company doesn't even replace the solder, they just heat up the solder already on the board, which I'm not really a fan of...
Hope this helps.