I acknowledge that this is a difficult issue to deal with.
That means the First Amendment just does exactly what it was intended to do.
I acknowledge that business can refuse service but its legality can be challenged such as in cases of:
1) Refusing service to customers who are identified as military.
http://www.snopes.com/politics/military/refused.asp
2) Refusing selling food to those who are identified as homeless (even when they have money to pay).
http://www.eater.com/2015/5/8/8573603/mcdonalds-refused-service-homeless-man-england
The employee claimed he did nothing wrong as he was complying to company's "zero tolerance policy on serving homeless people" (the company denied having such policy.)
3) Pharmacies refusing to dispense or transfer Prescribed Contraception:
http://nwlc.org/resources/pharmacy-refusals-101/
Six states allow such refusal practice citing religious or moral reasons.
Others including California prohibit such refusal.
4) Refusing service to those who are identified with a lazy eye: Not an actual case but a theoretical example cited by:
http://www.legalmatch.com/law-library/article/restaurants-right-to-refuse-service.html
I could make up another example:
A store owner recognized a customer who's also a friend and said:
"You know how much I've been supporting Donald Trump with all his signs and pictures in this fine establishment
and I am sorry, your internet blog calling him "a jerk" is certainly not good for my business.
You are free to do business with the one across the street but not here."
The store owner may invoke the right to refuse and cite: "It's business, not personal!"
However, by retaliating customers for posting offending internet blogs against a company's favorite political candidate, that company's action can undermine the freedom of speech because customers might have to start writing praises to a candidate they don't support so they can receive service.
So what all this has got to do with Musk and Alsop?
Musk tweeted: "denying service to a super rude customer"
The reason was due to a classification.
That classification is:
1) not military,
2) not homeless,
3) not prescribed contraception,
4) not lazy eye
5) not writing Trump as "a jerk",
But the classification is:
"super rude customer."
How do you classify Alsop as "super rude?"
Did any one report that he made a scene at the X Launch Event?
Did any body report that he gave Tesla a hard time by lingering around even when the event was well over very late in the night?
What we can see are Alsop's internet blogs.
Did Alsop criticize Donald Trump? No.
So whom did he criticize? Tesla CEO.
So how did Tesla respond?
Musk taught Alsop a lesson by using the company's ability to provide or withhold service and applying that negative reinforcement which might send a chilling message to a free speech community.
Is this a good business decision?
I am not sure!
When you cite a specific reason, you might have to defend that reason.
I've seen how Human Resource could fire a pregnant employee whom they did not realize that she was pregnant during the hiring.
They would not cite pregnancy as the reason for firing.
They would cite something vague such as "As agreed and signed by both parties, we exercise our right to release you from employment during a probationary period."
When you offer no reason, there's no need to defend nonexistent reason!
As to whether this "denying service" is legal or not?
I am no lawyer so just like the refusal to dispense contraception issue, courts might support or prohibit such practice depending on which states you are talking about.
In summary:
There are people who advocate the right to refuse service.
There are people who believe this has nothing to do with freedom of speech.
There are people who think it was a proper thing to do.
On the other hand, I have no legal background so you don't need to consider my opinion:
1) I believe the right to refuse service does have its limitations.
2) By publicly disclosing the reason as "rude" which can only be proven as "writing" on the virtual internet, not even in a real "physical" world, is an invitation for attracting First Amendment Scholars.