No, that's not what Turing complete means. Turing complete means it can do an arbitrary computation, but "implementing Rust" usually means it needs to be able to take in a string of code and produce a binary, which means your program needs to have some way of actually doing that. Sure, you can encode the compiler into the Turing machine, but an arbitrary Turing-complete tarpit may not actually have the syntax to know what a string is. Usually the best you can do is encode the programming language into some form the machine can understand. (For example, with Fractran you'd encode your input as some sort of Gödel numbering before giving it to the program.)