Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?
Yes, you can find such a proof in the following article:
Citation: Singh, D. (2006). The spectrum in a Banach algebra. The American Mathematical Monthly, 113(8), 756-758.
The paper is easily located behind a pay-wall (JSTOR) here.
Here is an image of the start:
As promised, the article concludes with a proof of Liouville's Theorem following from Theorem 1 here, that is, using the non-emptiness of $\sigma(a)$ as specified by the OP. Perhaps someone else can find a copy of this article that is freely accessible.