Does openly and aggressively berating or insulting people who voted for Hillary or Trump warrant an Article 134 violation under UCMJ?

Honestly, if you are not mature enough to understand that your candidate won or lost, then proceed to rub it on someone's face then you shouldn't be running your mouth.

Insulting or berating fellow service members for their political beliefs should not be tolerated. It takes away from the good order cohesiveness of your unit. I've been out of the military for almost 50 years so I'm not sure what an Article 134 is. I know when I was in, I didn't worry about politics or who the POTUS was. I just concentrated on doing my job and obeying my NCOs and officers.

IMO, yes that behavior is unacceptable regardless of who you are and more so for people wearing the uniform of our country. BTW, yes I include the POTUS in that statement.
THe UCMJ has a number of articles whose main purpose is to maintain good order and discipline and make it so our troops can function as a cohesive whole. Political backbiting has no place in that environment.
When I served back in the ancient days of the Reagan administration I could have cared less about politics other than if the CiC was placing us in harm's way and whether he was allowing us to stand proud and do our jobs. I can remember a few conversations with my mother who hated the man because of his domestic policies but at that time those did not matter to me as much as his foreign policy since it affected me more on a day to day basis while I served.

Perhaps this will help everyone out. "Public Affairs Guidance for Political Campaigns and Elections" document: http://tinyurl.com/d9ovwxh
I copied that link from a 2012 article on the army.mil site.

I, for one, do not want to hear anyone complain who does not vote. PERIOD! I have a roommate who does not vote, and I never hear a political complaint out of him what so ever. Though, I am a bit of a conspiracy theoriest, I still think we should try our best to work with this POTUS, and help him drane the swamp.

Article 134. General article:
Though not specifically mentioned in this chapter, all disorders and neglects to the prejudice of good order and discipline in the armed forces (being a pr*ck), all conduct of a nature to bring discredit upon the armed forces (being a partisan pr*ck), and crimes and offenses not capital, of which persons subject to this chapter may be guilty, shall be taken cognizance of by a general, special, or summary court-martial, according to the nature and degree of the offense, and shall be punished at the discretion of that court.

— 10 U.S.C. § 934

It is never correct to disrespect or be rude to people based on their political views or who they voted for. As an American we have the right to our own opinion be with political or otherwise. However, I do not agree with the statements that say military shouldn't have a political opinion and that we must respect who is the president regardless of whether we like them or not. Just as with all leaders in the military, respect is earned, not freely given. Yes, you need to respect their rank (That is good order and discipline). Yes, you need to follow their orders (as long as they are not illegal or unethical). But, there is nothing that requires you to blindly respect the person. I have never understood how we went from a country that thought of their military, police, firefighters, and medical as first rate citizens to a country that believes it is okay to kill police, shoot at firefighters or medical, and take the right of freedom away from those sworn to protect the country (i.e. military). Do we have a right to vote? Then we have a right to our own political opinion, views, and the right to discuss them amongst our friends and family. Just like I have the right to march in a parade! The only right I do not have is to affiliate the military with my political views (such as wearing my uniform during protests, or claiming that the military views one way or the other). My rights as an American don't stop just because I wear a uniform! I am simply held to a higher standard of discipline, and that means knowing when and where it is okay to express my political views.

As I'm answering after the second edit, probably not many will read this, but here it goes anyway. It could get to a 134 violation if the commander deems necessary and it violates good order and discipline. I don't foresee too many commanders pursuing this avenue, but it would be well within their authority to do so. I'd say NJP via Article 15 would be the most offered from a unit commander. Most likely it'll be a supervisory counseling statement or similar. I will disagree with only one point you made. Respect for the POTUS is not a given, respect for the office and position is. Just like officers, NCOs, etc. An enlisted person salutes the rank and position, not the person. The person could be a total piece of crap, which sometimes happens. Same goes with NCOs, subordinates respect the NCO by standing at parade rest, etc when they are talking. Same can be said of the enlisted corps, not everyone is awesome and there are some awful examples out there.

Personally? Never had much use for voters or their decisions. As a member of the military i NEVER VOTED simply because the result didn't matter to me, i still did my job.

Now as a civilian i do vote... i assume others vote their feelings just as i do... i just assume everyone else is wrong... :-) no problem!

I always felt that as a SM you still have all your rights just like a civilian. You just have to limit their use due to the restrictions placed on you by your service in the Armed Forces.