You want Peterson fired because a cutie pie account sent out a fake tweet, and you think that makes you mature. LOL. We are both dipshits for getting sucked in by it.
But even in the fantasy scenario where Peterson did actually send that tweet - LOL at thinking that wanting the victim of vicious slander to be fired for responding to said slander makes you 'mature'. Incredible logic.
In that situation, you can either give the allegations credibility by threatening to sue Cruz or demanding an apology, which gives the whole story more oxygen and a chance for Cruz to double down and make it worse. Or you could mock the allegations and the person making them, thus pouring cold water on it and making the allegation seem rightly ridiculous.
Now, it would be a risky defence in case there were some athletic commission official with the same idiotic reaction as yourself. But it would not be an immature way of defending yourself. Threatening to sue would be immature, and that's where lots of people immediately turn to when slandered.