Relation between "undecidability of arithmetic" and "godel's incompleteness theorem"?
Yes, if we know that $Th(\mathcal{N})$ is undecidable then we know that it is not computably axiomatizable, and in particular we know that no computably axiomatizable theory consisting only of true sentences of arithmetic is complete.

Here I'm adopting a Platonist view with regards to $\mathcal{N}$: I assume that "the" set of natural numbers exists, and "true" means "true in that structure.

However, there are a couple points here:


How do we know that $Th(\mathcal{N})$ is undecidable?
What about strengthenings of Godel's theorem, like "No consistent computably axiomatizable theory of arithmetic extending PA (or even Q) is complete"? (This is an extension due to Rosser of the theorem as originally proved by Godel, but is basically just one clever idea on top of the usual proof.)


Ultimately, (1) really already uses the key idea/argument of Godel's theorem, while (2) points out the limitations of measuring the complexity of a single structure.

So this shouldn't be construed as trivializing Godel's theorem in any way; in particular, I object to the word "simply" in the last line.