I read on https://www.20minutes.fr/politique/2676275-20191216-reforme-retraites-pourquoi-jean-paul-delevoye-demissionne (mirror) this tweet from Jean-Paul Delevoye @delevoye:
J'ai présenté, ce jour, ma démission au Président de la République et au Premier Ministre.
It is the first time that I read "ce jour" used in this way. Can "ce jour" and "aujourd'hui" be used interchangeably?
No, they can't if one goes by established usage and in at least one context it is not at all possible on the count of non idiomaticity; they do mean exactly the same thing in certain contexts but the use of "ce jour" is somewhat restricted in this sense that it tends to communicate a certain formalism that people associate with important happenings in the world of administration, justice, politics, in the context of formal declarations in the world of business, science, education, and in the case of solemn declarations for example. Although "ce jour" occurs in other expressions where it does not mean "aujourd'hui", this latter word, even though it has no other signification than "today", is much more common (ngram), and that is a clue to the fact that in the plain contexts of everyday speech it is used almost uniquely.
You can use "ce jour" in everyday language if you want to insist on something and that is again conferring to your utterance some formal touch or solemnity.
J'irai, ce jour, essayer d'arranger cette histoire, je vous le promet. (possible emphasis on "ce jour") (user LPH)
« Je l'accepte , ô mon père », répondit Alexandre, « et, à compter de ce jour, je me ferai appeler votre fils, si vous me donnez l'empire de la terre entière. » Le prophète rentra alors dans le sanctuaire, et reprenant la parole quand les hommes ...(from the literature, 1837)
Tes magnanimes projets m'ont rempli d'une telle admiration que, dès ce jour, je me déclare ton a vassal et ton esclave. (from the literature, 1824)
"Ce jour" is also used to refer to a given day that has been mentioned in the conversation; it can be a day in the past or in the future.
When preceded by the preposition « à » (à ce jour) this locution often means "aujourd'hui", but with the idea of this day defined as "aujourd'hui" being a limit (meaning: "since a certain event" (that the context might make clear) "until the present day" or simply "from some indefinite time in the past until today"). It could be parahrased in French as "depuis ce temps-là jusqu'à aujourd'hui" or simply "jusqu'à aujourd'hui". "Ce jour" cannot be replaced by "aujourd'hui"; you'd have to say "jusqu'à aujourd'hui", "à aujourd'hui" would be incorrect. Thus, here is a case when an exact substitution is not possible. The idea of the event in the past might not be clear from the context or there might be no such event.
This ngram might induce one to think that "jusqu'à aujourd'hui" is not as popular in this linguistic context.
À ce jour, je n'ai pas eu un seul client qui se soit plaint. (You are then given to understand that the whole business life of the person is the period of time implied or for instance that the period of time for which a given item has been commercialised is what is meant.).
Ce que je peux dire c'est qu'à ce jour, je me méfie plus des produits OGM que de la viande. Ne me demandez pas pourquoi, c'est du feeling et en général, si je l'écoute, je ne me trompe pas. A chacun de sentir ce qu'il en est pour lui ou elle. (from the literature)
