Realism is one specific path towards immersion. It's neither a necessary condition of immersion, nor is a game's level of realism at all correlated with how immersive an experience it can provide. Realism is nothing more than a game's resemblance to real life. Real life has an open, impossible to fully articulate (as far as we've been able to, anyway) set of rules, while games have their own sets that are generally self-contained, fully-definable, and self-sufficient. When we're immersed in a game world, it's not because it is real, though strong realism can aid in immersion; we become immersed because we buy into the systems and metaphors of the game. This buy-in requires that the systems and metaphors be smooth to our mental touch. Awkward metaphors, obvious technical issues, and broken game systems can open gaps in the closed system and force us out of buy-in. Other activities outside of the game itself can also hinder buy-in, too, like a crying child, feeling ill, a headache, or just being in a bad mood. When immersed, you and the game are communicating smoothly. Any break in that communication or unwillingness on either side to communicate has a significant chance of breaking immersion.
Game design is communicating interesting problems to the player and then giving him the tools to communicate back solutions that the game then somehow tests and gives feedback on. Game mechanics are communicated through metaphors that reference fantastical or conceivably real objects through the simulacra of sprites, models, textures, and sound. The most obvious way to communicate with a player is to use a "language", or set of metaphors, that they already know: such a language is how-the-real-world-generally-works. This is a shortcut to immersion. Of course no game is truly realistic, but we don't mind that because the exceptions to realism in an immersive game are generally mechanically and metaphorically consistent and make the gameplay better. Games that don't aim to be realistic still use the real world as a basis for the metaphors that pull the player into the game. Realism isn't necessary for immersion, but the game does need to provide the player with ways to relate to the game world.
All games have a base level of likeness to the real world. Realism beyond this point has no correlation to the game's ability to provide an immersive experience.