is the voting process considered to be the ultimate arbiter of such disagreements?
I'd say the reader is the ultimate arbiter -- technically there's no need for any consensus such that "the community stands behind answer X". Presumably the person who wrote the question stands behind the accepted answer, and people who write answers stand behind them. Neither of which is any guarantee of accuracy.
Votes are a much better indicator, but it is still not impossible for an upvoted answer to be wrong, and it is not hard to find examples where the most upvoted answer to a question is far from the best one; this can happen with questions where there is a lot of voting in a short time, and better answers were added later.
It feels a bit awkward and at cross-purposes to be critical of another's answer when you believe your own answer is the correct one.
If you feel this way, you should make it clear. If you run across any answer you think is wrong and you can make the point succinctly, do so, even if you are not leaving another answer.
I'd recommend avoiding too much back and forth. While it would be great if everyone recognized the error of their ways when such is pointed out to them in reasonable and objective terms, this is not the always the case. At that point, it is not necessary for you to convince the author -- it's more important you convince the audience, and bickering probably isn't effective that way. If you have a valid point and you make it plainly, that's enough. Either it hits home or it doesn't.
Conversely, if someone leaves a comment on a post of yours claiming you are wrong but you think they are wrong, don't feel obliged to engage them if you don't think anything constructive is going to happen. If you want to update your answer to clarify this, do so, and perhaps leave a reply indicating you have addressed the criticism.
"false facts" seem counter to the objectives of this forum.
A nice thing about technical topics is that there usually is an objective right and wrong that most practitioners can come to a clear consensus on. This may require some fundamental knowledge on behalf of the reader, but that is not really your responsibility. You do not have to make everyone in the world understand the nuances of protocol X in 100 words or less. If you can make enough people to whom the issue might be concretely relevant (i.e., people who are looking for and can understand the answer) understand, great.