Also, one certainly can say there is not contradictory messages in this text or all of Scripture! You took the John 7:24 text out of context. The text says:

“Do not judge by appearances, but judge with right judgment”

So the right kind of judgement is one that does not judge based on appearances. The Pharisees (Note the “Jews” referenced in John are religious leaders contextually) judged many people to be sinners and themselves righteous, yet the Pharisees were deserving of greater condemnation(Mark 12:40). Therefore we judge rightly when we recognized we are all in the same hell-bound boat apart from Christ. We need divine mercy! See Jesus’ parable about the Pharisee and the Tax Collector. The Tax Collector walked away justified even though he was judged by the Pharisee to be a sinner, yet the Pharisee if he judged rightly would know that he needed the same chest-pounding repentant grief over his sin (Luke 18:1-8).

Rom 12:10
as to brotherly love, kindly affectioned towards one another: as to honour, each taking the lead in paying it to the other:

Rom 12:16
Have the same respect one for another, not minding high things, but going along with the lowly: be not wise in your own eyes:

Rom 14:13 (ESV)
Therefore let us not pass judgment on one another any longer, but rather decide never to put a stumbling block or hindrance in the way of a brother.

Rom 15:5 (ESV)
May the God of endurance and encouragement grant you to live in such harmony with one another, in accord with Christ Jesus,

Rom 15:14 (ESV)
I myself am satisfied about you, my brothers, that you yourselves are full of goodness, filled with all knowledge and able to instruct one another.

1Cor 12:25 (ESV)
that there may be no division in the body, but that the members may have the same care for one another.

1Thess 4:18 (ESV)
Therefore encourage one another with these words.

1Thess 5:11 Therefore encourage one another and build one another up, just as you are doing. ESV

1Thess 5:15 See that no one repays anyone evil for evil, but always seek to do good to one another and to everyone

1Pet 1:22 (ESV)
Having purified your souls by your obedience to the truth for a sincere brotherly love, love one another earnestly from a pure heart,

Jas 4:11 (ESV)
Do not speak evil against one another, brothers. The one who speaks against a brother or judges his brother, speaks evil against the law and judges the law. But if you judge the law, you are not a doer of the law but a judge.

Jas 5:9 (ESV)
Do not grumble against one another, brothers, so that you may not be judged; behold, the Judge is standing at the door.

Jas 5:16 (ESV)
Therefore, confess your sins to one another and pray for one another, that you may be healed. The prayer of a righteous person has great power as it is working.

and so on…

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I’m not saying it’s easy to figure out how to follow Jesus when in one place he commands, “Judge not,” and in another place he instructs that we “judge righteous judgment.” I’m just saying you can’t pretend that there aren’t any contradictory messages.

I’m not so interested in Jonathan Edwards’ personal opinions on this matter; unless you’re trying to argue that he’s a prophet or in some other way infallible on matters of theology, he’s just another guy with his own interpretation. Nevertheless, I’d say that in this passage you quote he endorses “speak[ing] of [others' problems] with grief and pity.” Do you think that would not still be embarrassing, or disapproving? Would that not constitute pointing out another’s failings?

Short answer: In a church the only morally legitimate judgments of people are those exercised by the elders in formal church discipline, a process strictly defined/limited by the Bible. What we don’t need are vigilante church factions asserting whatever judgments they feel are right.

Not-as-short answer: Jonathan Edwards helps us locate ourselves where we all need to be when he clarifies spiritual humility, slightly edited here:

“Spiritual pride is the main door by which the devil comes into the hearts of those who are zealous for the advancement of Christianity. It is the chief inlet of smoke from the bottomless pit, to darken the mind and mislead the judgment. It is the main source of all the mischief the devil introduces, to clog and hinder a work of God.

Spiritual pride tends to speak of other persons’ sins with bitterness or with laughter and levity and an air of contempt. But pure Christian humility rather tends either to be silent about these problems or to speak of them with grief and pity. Spiritual pride is very apt to suspect others, but a humble Christian is most guarded about himself. He is as suspicious of nothing in the world as he is of his own heart. The proud person is apt to find fault with other believers, that they are low in grace, and to be much in observing how cold and dead they are and to be quick to note their deficiencies. But the humble Christian has so much to do at home and sees so much evil in his own heart and is so concerned about it that he is not apt to be very busy with other hearts. He is apt to esteem others better than himself.

Some who have pride mixed in with a heightened awareness of God’s glory and intense experiences of spiritual joy are apt to rebuke other Christians around them for being so cold and lifeless. But the humble, in their joys, are also wounded with a sense of their own vileness. When they have high visions of God’s glory, they also see their own sinfulness. And though they speak to others earnestly, it is in confession of their own sins. And if they exhort other Christians, they do so in a charitable manner. Pure Christian humility disposes a person to take notice of everything that is good in others and to make the best of it and to diminish their failings.”