Departmental or school tenure and promotion committees review a dossier of letters of reference, a CV, a personal statement, and perhaps other material. They present those findings to the department as a whole for a vote. Perhaps you believe their interpretation of the dossier is incorrect, whether it be denial or a tenure/promotion.
1. If you are on the committee, you might file a dissenting report. Or you might argue later on, whether in the faculty meeting or in letters to the higher-ups.
a. It does you little good to argue against a particular letter writer, as such, since you chose the letter writers.
b. So, what you do is to make a list of the main points brought out in the letters and in the committee report. You then present your argument in terms of those points. No ad hominem of the committee or letter writers. Rather, you will be presenting your position on those points, with whatever authority you possess. If there is counter-evidence, of course you want to present it. What you are trying to do is to make people think twice.
c. Don't distort the letters or whatever else you are referring to. Selective quotation, insults to the letter writer, etc. are all likely to have your position weakened.
d. Say you are the strongest person in the field in your department. Your judgment might well be given greater weight, but still you are making an argument. Your opinion, as such, may well be important, but it is vital that your argument be fair, to the point. If you do not believe the candidate's methods are appropriate, say why....
2. If you are not on the committee, you are welcome to question the credibility of the letter writers. But you cannot be seen as biased or too selective. Credibility is questionable if the writer is too closely allied to the candidate, if methodological preferences seem to sway the letter writer, etc. Still you are likely to find yourself acting as in #1 above.
3. If you are in a higher-up position--chair, dean, university committee--you are welcome to act as in #2 if there is good evidence. You may well have received a memo from a dissenting member of the faculty. But it is best to first read the dossier of that is presented by the committee, and then the later material.
4. What you are trying to do is to sway your colleagues. But you are also building into the record your dissent, and in so far as your dissent is fairly and accurately argued it will more likely be effective.
5. In the end, the university will survive.
If you believe the candidate should not have been tenured/promoted and the university decides otherwise, make sure the candidate gets the kind of mentoring that will make them more worthy of your good judgment.
If you believe the candidate should have been tenured/promoted but the university decides otherwise, help the candidate find a good position elsewhere. That you disagreed might well be confidential information, unless the dossier and related materials are to be public. That you believe the candidate is worthy is something you can share with the candidate as you help them go forward.
6. People often carry grudges for years, either because they have "lost" in this process. I am not talking about the candidate, rather the faculty and the committee. Again, the university will survive. And you have better things to do. If you don't, find another job.
7. As for legal remedies, I have no good advice.