Anyone know what Army Regulation (or any branch) that allows a higher-ranking officer to call a lower-ranking officers by their first name?

You know as well as we all do that all branches of the military can go a bit overboard with regulations. That being said, I simply can’t imagine someone writing a regulation to the effect of “Under the following conditions, senior personnel are allowed to address subordinates by their given name (then goes on to list 7,000 conditions).
I think someone writing that would set his/her hair on fire, then put it out with a hammer.

Anyone who has a problem with this issue is probably just looking for an issue to have a problem with.
Of all the names one could be called, a first name should be the least of their concerns.
I've always felt respected when a senior officer used my first name, and on the instances where I called a Soldier by their first name, I used it respectfully.
I'm sure that there may be exceptions with other people's experiences, but those times, most likely, had extenuating circumstances involving personality conflicts, their performance, or both.

Both of my children are members of the Millennial generation. My son is an Iraqi Vet. As a Vietnam Vet and a single mother, I was strict with my children and I recall participating in a conversation with my peers who were discussing their children's behavior, all of our children were teenagers at the time. I stated that when my children were little I spanked them when they misbehaved (even though Congress passed a law making spanking illegal) and that I took my children to church every Sunday. A line I frequently used when they misbehaved was, "How do you think God feels about what you have done"? Today, both of my children are married with children of their own, they both have excellent jobs, have great homes and both seem to be well adjusted and reasonably happy and their spouses don't complain so I assume they too are reasonably happy. I also want to state here that you will not find any weapons in either of my children's homes. My point here is: I taught my children to have respect, for their elders, for each other, and for those who have earned the right to outrank them. They understand that just because someone outranks you does not mean that person is superior to you or better than you but 'respect' is 'respect' regardless of Tina Turner's song. If a society has no structure it will soon collapse. There are far too many Millennials 'wandering' because their parents and/or grandparents failed to do their jobs as parents and "parent". If you bring a child into this world it is your job to 'parent' that child.

I remember on rare occasions the Chiefs in the Navy calling each other by their First Names in semi-private conversations. This was generally looked at as a sign of respect and familiarity amongst the senior enlisted that they . A privilege that was reserved for them.

Always think it's weird at first, but if the person being to that way doesn't have an issue, then no big deal. I just hate it if somebody corrects enlisted or NCO's referring to their subordinates or peers in the same manner.

Here's a place to start https://en.wikipedia.org/wiki/Uniform_Code_of_Military_Justice also there are additional links at the bottom of the linked page itself . So with the ability to read and understand, the only thing standing between you and your answer would be some work to find it . Remember the UCMJ is the definitive legal basis for all service members , here say has never been advisable or ad missible ( in other words If I tell you whatever and your dumb enough to take my word for it then your dumb enough to take the punishment as well ) . It's always best to have a complete understanding of the actual authority that you willingly signed yourself up for ( we have a volunteer armed forces networks as the draft has not been currently initiated ).

Our CO knew just about everyone on our ship by first name. But a destroyer is a small community and our CO was a mustang!

Hey, CW4....I was only an E5 in 1967 (SP5--E5---after 18 months of active duty..).BUT...I cannot understand how an intelligent adult in the USA (service or not) can ask such a question. I made E-5 in 18 months--I worked hard for that---in that period, I might have called a superior( E- or O)- by their first names 30 or 40 times (E-90%)-----I was called by my first name.....maybe 300 to 500 times...if a "superior" "officer" does not understand common courtesy...and the privilege of "command"...perhaps they should not be a "superior officer"

SP5--E5---John Coggeshall---US 51-536-582--

I think it is a superior officer’s way of showing familiarity and affinity for a subordinate while also reminding them of the rank structure.

This has been a complaint of mine. I understand why they do it but I feel it is unprofessional. I also feel if they do it in front of senior nco's they may slip up and do it in front of the enlisted. 1SG Wheaton