I think perhaps Danforth is a coward. If you were to ask him, he would say his job is upholding the law despite any objections and pressures. He is in this position of power and it's clear he doesn't like to be questioned or doubted. When he does have a doubt, he's unwilling (afraid) to pursue it.
The judges and the girls ate and socialized with each other, to at least a small degree, outside of the courtroom. A man of discernement should have been more alert to the duplicity and dissembling of the ringleader, Abigail. The moment I think he shows at least one cowardly bone is when Abigail delivers a veiled threat to him, letting him know his may be the next name she calls out. That should have been a clue--if she would make up a lie about him, she was capable of doing so about others. Instead, he carries on blindly and persistently because he wants to and he can.
While it's true he's stubborn and arrogant (as we see in all his dealings with Proctor and especially when he's eliciting Proctor's confession), he's also rather cowardly. He's afraid to find out the truth--that his lofty and important work on this court was all for naught. Even worse, it was all a lie.
I do not think that Danforth is a coward. I would define a coward as someone who knows what the right thing to do is, but does not do it because they are afraid. I think that Danforth is selfish, not a coward.
To me, Danforth thinks that whatever is good for him and the offices he holds is the right thing to do. He cares most about holding on to power and credibility. So I don't think that he is saying to himself that he knows the court is wrong. I think he believes that it is right because it helps him hold on to power.