If we think of self-awareness as an evolved brain circuit phenomenon, its existence probably has little dependence on language in particular. This is evidenced by the fact that Alzheimer patients loss of self-awareness, as well as changes in self-awareness due to injury are associated with the frontal lobe. That's not to say that the self and self-awareness aren't heavily modulated by language (as are most mental constructs!) There are several non-human animals that pass the self-awareness test but most of them are creatures that many consider to be closer to the intelligence of humans than other animals (namely, great apes, elephants, and dolphins). Consequentially, many of these animals have also developed methods of communication, though language might be too strong of a word.
Finally, such a dependence would imply that the strong version of the Sapir-Whorf hypothesis has some legitimacy (especially when it comes to the self) but there's been no evidence found in support of the strong version, and most cognitive scientists favor the weak version of the Sapir-Whorf hypothesis.