What a strong negative connotation! Judging. Judging things, judging people, being judgmental... these are not pleasant things that polite and civilized people do... right?
But if judging is such a fault, perhaps we should learn what exactly it is. Hmm... I don't know exactly. Maybe Merriam-Webster can enlighten us:
Judge. Transititive verb. "to form an opinion about through careful weighing of evidence and testing of premises"
Hmm. That's... not what we expected, now is it? Judging is bad. But forming an opinion through careful weighing of evidence and testing premises... that doesn't sound so bad. Hmm.
What does that sound like to me?
Evaluate. Transitive verb. "to determine the significance, worth, or condition of, usually by careful appraisal and study
It sounds to me like when people having a problem with "judging," they're disliking someone determining the significance, worth, or condition of something when they shouldn't be...
Who are you to evaluate?!
That doesn't... sound... quite... right...
Our little expedition here suggests that someone who calls out, "Who are you to judge?!" is expressing strong disapproval at someone for evaluating something.
What's the opposite of being judgmental, then?
Refraining from judgment is refraining from evaluation. Not judging, then, is not forming an opinion.
This is remarkably difficult to do. We naturally form opinions and judgments. It seems to me, then, that the easiest way to refrain from judging is curiosity. When confronted with an opinion that's foreign to your understanding, you would say, "That's interesting, why do you think that?"
It strikes me that many people who would label themselves "nonjudgmental" are similar to the conquistadors who came in peace. It's like, "Hi, we're here in peace. Now swear allegiance to our God or we're throwing you off this cliff."
Someone who says, "How dare he be so judgmental! What an asshole!" is... well, it's ironic, eh?
Judging is evaluating. Someone saying, "How dare he judge!" is saying, "How dare he evaluate this topic in this way!" Many (most?) supposedly tolerant people are tolerant of everything except disagreement with them. True nonjudgment is curiosity, and trying to understand where another person's opinions come from.
Condemning someone for evaluating - evaluating differently than you, that is - well, it's an odd thing, isn't it? "We're here in peace. Now we're putting you to the sword if you don't do what we want." It's getting so caught up in ideology that you believe your own conduct is righteous, and not holding yourself to the same standard you demand.
(I'd advise against doing this, because my evaluation is that it's conducive to ignorance.)
But who are YOU to judge?
Indeed, there are plenty of people who make evaluations without the necessary skill and knowledge to make correct evaluations.
But in that case, you'd be much better off pointing out the error the person is making instead of bringing out the (hypocritical) moral indignation and attempts to shame people for having different opinions than you.
True tolerance is curiosity. But making evaluations and coming to opinions based on evidence, testing, and refining - this is healthy too. To make a significant difference in the world, you need to come to your own evaluations. Some of them will be wrong, and then you'll want to refine your evaluations with new evidence.
As for the people making loud moral condemnations? Feel free to link 'em here. If you did just get linked here, well, hi there. Look, I'm not against judging - evaluating and facts and coming to your own opinion - but I do think it's kind of silly to judge someone and try to shame them for evaluating and coming to their opinion. If you're truly interested in being nonjudgmental though, try curiosity - it's a marvelous way to suspend judgment.