Are patents appropriate in taxpayer funded academic settings? I am asking because I work for a non-for-profit academic hospital where the vast majority of the funding comes, one way or another, from the taxpayer. It is certainly possible, actually encouraged, for us to file for patents should we find some new way to diagnose or predict disease progression.

But, I will mention this again, we are funded by the taxpayer so we are already getting a salary and resources to do our work from them. Is it filing a patent and consequently delaying the publication of our research in order to find out a way to commercialise the work something that could be considered ethical?
Yes it is ethical.

The patent will typically be owned by the institute and it will get most of the revenues (this is why they encourage it). The inventors usually get some form of compensation. Revenues from commercialization is partially how your salaries can be payed and could in fact save tax-payers money.

More importantly, developing a new diagnostic or therapy will in the end improve the lives of the general public (and even save lives in many cases). Currently, due to the immense costs involved, the only way to get a new therapy to the public is in a commercial setting.