A doctor, who was also an English grammarian and wanted to be entirely clear, would likely prefer to say only that you are immune to the pathogen he immunized you with.
Saying immune from has the possibility of implying that you are immune because of the pathogen. Though it is a weakened form of the pathogen that made you this way, it is still not those poor little, long dead guys that the doctor is trying to convey you are immune to, it is all the other, (and full-strength) pathogens of that particular disease that you are indeed immune to. In fact this very doctor might have a situational need to say that you are indeed immune from those poor little, long dead guys that he injected into you, hence another reason to be strict.
This, of course, is a parsing of the literal meaning, but the idiomatic use is derived directly from the literal and does not have currency on its own, without its etymological origins in the medicinal use.
Now, immune against is just wrong because of either a seeming redundancy or plain misuse.
In the case of calling it redundancy I mean to point out that the quality of being against something is built into the word immune, so the against in, immune against, is superfluous, somehow.
In the case for just calling it misuse I think it is odd to be against something that, in actuality, cannot harm you whatsoever (as a result of the ostensible immunity to it).
Immune to strikes as being the only reasonable usage for your example, assuming that the statement means to say that the president cannot be harmed by criticism because of his position.
I'd leave open the possibility that, like in the doctor's example, that a person could become immune from criticism. For example:
"Her life growing up with four brothers left her immune from criticism, hence the booing from the audience did not faze her."