You are correct that technically you can be fired for what you say, but not when it's protected speech.
You would not support some one being fired for voicing political opinions that you agree with, so this is not only illiberal but also quite likely hypocritical.
In some states in the US, you can't legally fire people for lawful off-duty activity, including political activity. There are some states where political activity specifically is protected. But, there are plenty of places where you can be fired for any reason.
In many cases where off-duty activity is protected, there are exceptions for when that activity conflicts, or (sometimes) is seen to conflict, with your employer's interests.
Even in cases where your political activities are protected and no conflict can reasonably be inferred, you may be in a "at will" jurisdiction that allows your employer to fire you for no reason, which then puts the onus on you to prove to a court that the real reason was because of your protected activities.
While some states do not provide protection for political activity at the workplace it doesn't mean that discriminatory actions for political activity are legal, in some states there are also specific protections for election years.
And yeah the whole US "at will" thing is an issue on it's own, the fact that this is the law of the land is pretty absurd to me.
And, as an extension, the beliefs of it's members should never be taken to be the beliefs of an organization?
A public figure == https://en.wikipedia.org/wiki/Public_figure
Should they be fired for attending a similar rally against gay marriage?
I don't see how being a public figure means you don't have the right of protected speech.
But you know I'll bite; your link states that you can become a involuntary public figure, you don't seek it but you still are and you are considered as one for this case.
Let's say the teacher was picked up by a news crew and made into a public figure.
Can they be fired then?
The fact that some how it can or should spill onto to your employer just shows how fixated people are about partisanship.
If I express a political opinion under a clear mandated of protected speech like during say a political rally there is nothing my employer should be able to do to me. This is something we must protect as its critical to democracy and pluralism.
Unless you are dandy with how we used to fire commies in the 60s and 70s.
It amazes me that people can consider them selves being liberal and don't understand the concept of protected speech and free speech.
They can require you not to do that at the office, or using any official company channels (as long as this policy applies equally) but if you comply you should be protected from any sanctions.
Without this pluralism dies and with it democracy, and this is how you get partisanship to its current levels.