On the one hand, yes, it's possible; it's how routers work in the first place – by routing IP packets between computers with different addresses.
But on the other hand, it's not going to work, because most likely your ISP only assigns you one global (aka external) IPv4 address. (Partly because IPv4 addresses are in very short supply – the internet is only slowly moving to IPv6 – and partly because ISPs like to charge extra for things like extra IP addresses.)
(The reason you can connect multiple computers to your router/gateway is because the router has a NAT function ("address translation"). With NAT enabled, it assigns "local" (or internal) addresses to all connected computers, and the "global" ISP-assigned address doesn't belong to any of your computers – it belongs to the router. The NAT function rewrites every IP packet to make it look as if all computers are using the same "global" address rather than their own local ones. This is also the reason many games and P2P programs ask you to set up "port forwardings".)
If you call your ISP and ask them for a second IP address to be added on your connection – which usually costs a bit extra – then it's of course possible to reserve a specific global address for a specific computer. (Though there's the question of how configurable your router is; some devices can achieve this in several different ways, and some other "home gateways" have just the bare minimum of settings.)
Unless you use a proxy server the only way to do this would be to get another internet connection. – Matthew Williams – 2014-04-23T13:11:04.590
You ALREADY have different INTERNAL ip addresses. So you don't mean IP addresses generally you mean strictly, EXTERNAL IP addresses. I edited your question to say EXTERNAL, particularly as both answers interpret you(i'm sure correctly) as meaning external. – barlop – 2014-04-23T13:26:04.840