Are there any proofs the undecidability of the halting problem that does not depend on self-referencing or diagonalization ?
Yes, there are such proofs in computability theory (a.k.a. recursion theory).

You can first show that the halting problem (the set $0'$) can be used to compute a set $G\subseteq\mathbb N$ that is 1-generic meaning that in a sense each $\Sigma^0_1$ fact about $G$ is decided by a finite prefix of $G$. Then it is easy to prove that such a set $G$ cannot be computable (i.e., decidable). 

We could replace 1-generic here by 1-random, i.e., Martin-Löf random, for the same effect. This uses the Jockusch-Soare Low Basis Theorem. 

(Warning: one might consider just showing that $0&#39;$ computes Chaitin's $\Omega$, which is 1-random, but here we have to be careful about whether the proof that $\Omega$ is 1-random relies on the halting problem being undecidable! Therefore it's safer to just use the Low Basis Theorem).