Roy Moore is far more likely to be the nominee of the Constitution Party in 2020 than he is to mount a primary challenge to Trump. He's been mentioned before as the Constitution Party's nominee before, and he'd certainly give gravitas to the Constitution Party if he ran. He's a guy who might actually carry a few states, but who would ultimately lose big.
Moore shares some of Trump's constituency, but he's not like Trump. Trump doesn't bother himself with the kind of deep philosophical issues of Natural Law and Nature's God that Moore has made a career over. Moore speaks a language that I doubt Trump cares about; the language of principle, and while one may not agree with Moore's principled, one cannot argue Moore's character. I would also suspect that Moore's constituency involves many of the Ted Cruz supporters; it was CRUZ, and not Trump that was the favorite of REGULAR churchgoers in 2016.
If Moore actually challenged Trump, he wouldn't win, but he'd get a decent number of religious conservatives to support him. Moore's entry into the 2020 GOP Presidential primaries is the sort of event that might cause others to jump in and splinter the race. None of this is likely, however. I predict Moore will focus on his own re-election, and I believe he'll be re-elected easily.
I guess multiple someones have argued Moore's character. Rather successfully, at least to date.
I also don't think Moore will have re-election worries. Even given that it's Alabama, I believe that Jones will win the special election. I don't believe Moore will ever be elected to any public office again.