Apple confirmed that Game Center support would be arriving with the next version of OS X, Mountain Lion.
The company first demonstrated Game Center on Macs during the Mountain Lion preview, but today's press conference reconfirmed that the service would support multiplayer gaming between iOS and Mac gamers.
Since Game Center uses Apple ID, there's no need to create a Mac-specific log-in. If you've already created a Game Center ID on iOS devices, that same log-in information will work on Macs. Your friends list and achievements should also be ported over.
OS X Mountain Lion is planned for release sometime this summer.