Well, that might work, but that's the reason you need a router. Switches don't have DCHP, which is the system that routers use to give unique IP addresses to all of the devices connected to it. With IP addresses, the router can then manage multiple connections. With a switch, nothing is there to assign IP addresses, so the ports are all just there, and the first device to connect (at least, that's what I'd think would happen) snatches up the modem's one IP address and makes a one-to-one connection. I guess if that were the case, releasing the IP from the computer might work (try it and see), but I'd still recommend buying a router.
I have a modem, a router, and a 5 port switch, and the system works great to supply all my old computers hooked together in one wired/wireless network.