I think several things may have made you feel the sentence did not sound right:
First and foremost, what follows "so" in those constructions has to be an auxiliary or a verb like "be," which does not require an auxiliary. In traditional British English, where "have/has" do not require an auxiliary in the present simple tense, their inversion will be acceptable.
This is taken from Swan's "Practical English Usage" (3rd edition):
Therefore, in the present simple tense "do/does" are always correct after "so," and "has/have" (as main verbs) are only partially acceptable in British English (notice the example I have a headache. So have I above.)
Another thing that might be creating a sense of unbalance -- though this may be subject to different opinions -- is the plural of "actions" and the singular of "inaction." In a structure like the one at issue, I'd prefer plural-plural, or singular-singular. "Action" can be used as a non-count noun, synonymous with "acting," but "actions" sounds better. "Inaction" is only a non-count noun. Then, the best solution would perhaps be to choose a pair of plural opposites, like "actions and omissions." These two words appear as antonyms on many very well-written pages on the Internet, like this one: https://pages.stolaf.edu/ein/themes/acts-and-omissions/
Finally, the opposites already show contrast, so I think the use of "but" may be redundant and inappropriate. I'd say:
- Actions have consequences, and so do omissions.