TLDR: The Problem is You.
That's a strong expression, so I'll explain.
I am an adjunct in a reputable North American university. It is my first-semester teaching an upper-year/beginning graduate level course.
It's a reputable university. You're a beginner at teaching. Beginners in anything make mistakes. It's more likely you made mistakes than that the reputable university doesn't know its business.
A student raised a complaint against me to the chair of the department because they did not get the grade that they wanted. The complaint is without merit whatsoever as far as the student's coursework is concerned, however, they did mention that the distribution of the course grade was "lower than all the other courses he has taken" (obviously reeking entitlements here). The chair took issue with me solely on this.
And you should have taken that criticism as a signal that there was a problem with your approach to either teaching or marking (or both). You do not seem open to criticism directed at yourself.
The distribution was around 15% A, 30% B, 40% C and rest D. The average of the course was 77% which is a C in this particular school. The chair apparently saw this as a big issue.
You're an employee, so if your bosses say it's a big issue, then you should treat it as a big issue.
The compromise
Let's remember that word.
which we arrived at is to raise the grades of the student collectively.
You don't have to like this. It's a compromise and your employers want it that way. Just do it. Compromising means you should consider the matter closed and move on mentally, which you do not seem willing to do (not from the tone of your post).
The reasoning (or excuse) for this grade adjustment is that the students had poor preparations coming into a tough course. But the actual reason (which the chair mentioned) is because poor grade signals difficulty and hurts enrolment and graduation rate to the college and hurts my career because apparently students will avoid taking my course.
This is known as being practical and living in the real world. Those are legitimate concerns. You are in the business of education and you need to accept those realities.
The distribution of the grades for each professor is somehow available to the students which is a travesty in my opinion.
A different issue, but if you're not comfortable with that you can move on to a different job. It's how they do it where you work and you're unlikely to change that.
I am utterly at a loss. The grade adjustment is utter nonsense. I am moving students who cannot even multiply two vectors together into the C or B range.
Honestly it's hard to find that claim realistic at graduate level.
The school is highly reputable. But yet it seems that the chair is encouraging me to abandon basic ethics in favor of business.
It could simply be that you're not a match for the institute. Again, consider if you're suited to this role and/or this institute. Consider if the problem is you or them.
Worse yet, the student made up some completely fabricated lies against me to the chair even including that I have discriminated against them based on race/ethnicity (I am of the same race/ethnicity as the student) and the chair is not willing to say a word to the student.
Unless this had negative consequences for you (with respect to your employer), you need to mentally move on. In your career it's unlikely you'll avoid people making negative claims about you. Get used to it and develop a thick skin. If this did have a negative impact with your relationship to your employer you may need to move on or find a way to address the problem constructively.
Note that, if your tone in this question was anything to go by, it's easy to imagine you speaking down to students or coming across as arrogant or dismissive of their concerns. If you had e.g. a student who felt they had (elsewhere in the past) been a victim of discrimination, it would (again) be easy to see them interpreting a "teacher with an attitude" as "discrimination".
This may be a sign you need to work on how you communicate and interact with people.
I wonder if anyone else has faced this "pressure to raise grade" situation before and how you have dealt with it or rationalized it.
No rationale required. Your bosses made a decision and your job is to implement it. Period.
The real problem here is that you're not willing to accept what you describe as a "compromise" or let go of your personal anger. Nor do you seem prepared to accept your role in this could be the main issue.
No one else can solve those issues and you won't change everyone else.