Not that his billion followers should get their hopes up. Twitter isn't suited to nuance, and style often takes a back seat. Besides, Piers Morgan has probably already claimed copyright on the '#infallible' hashtag.
But what else could the Pope do?
Time and tide waits for no man. Imperative now demands that everyone gets with the technological programme. A few Popes further on and no doubt papal encyclicals will be sent out by text and possible sins flagged with a thumbs-down button on Facebook. Heaven knows what John Charles McQuaid would have made of it all.
The late Archbishop of Dublin was, in his day, a connoisseur of the more leisurely lost art of letter writing, and a collection of his ample correspondence now snoozes in quiet retirement in an archive in Drumcondra, visited only by academics. Thankfully, those of us without the excuse of scholarly research can now get a flavour of his preoccupations from a newly published selection of his letters. Though what's most noticeable about His Grace Is Displeased is how little seems to have changed.
Here he is, debating the location and ethos of a new children's hospital in Dublin. Here he is, looking into the condition of republican prisoners in jail. The book even begins with his thoughts on the Constitution, which, the archbishop's detractors will not be surprised to hear, was nowhere near Catholic enough for his tastes.
With a slight twist, these topics could all have been headline news from the last few weeks. The archbishop even gets involved in an entertaining spat about what sports are and are not appropriate for women. A phenomenon like Katie Taylor would have horrified him.
But if much of this material still feels like unfinished business, there's still one striking difference between then and now. Half this book is composed not of John Charles McQuaid's letters to other people, but of their letters to him.
Indeed, he once expressed frustration at the amount of correspondence he had to deal with. Everyone expected help with their problems, and, more inconveniently, they expected results.
In the chapter on censorship, for example, it quickly becomes apparent that ordinary people were just as outraged as the hierarchy about the spread of lewd films and books in Ireland -- including some who "consider the unfortunate Late Late Show as the source of all evils" -- and they expected His Grace to put a stop to it forthwith.
One man called round in person with an advertisement from the Evening Press which he wished to be "discontinued" after previous appeals to the newspaper and the Taoiseach had fallen on deaf ears.
The archbishop's preferred method of dealing with such requests was to have "educated ladies" approach proprietors and advertisers to express their concerns.
He had less luck when complaining directly to Basil Clancy, editor of Hibernia, after a Protestant theologian was invited to review a book of Catholic theology; but he was at least listened to.
That's the other big difference between the Ireland of the Fifties and Sixties and now. Not only do fewer people seek the advice and intervention of the clergy in matters of public interest, they're disinclined to listen when they hear such views anyway. The modern hierarchy can only dream of being indulged so willingly.
Last week when the church finally took a stand on the children's referendum, the statement went practically unnoticed, despite the fact that they did what everyone had been demanding of them by backing a 'Yes' vote.
That's how it is now. If they'd urged a 'No' vote, they'd have been vilified for continuing previous failings on child protection, but in coming down on the side of Yes, the church got no credit for it anyway.
We've gone from a position, under John Charles McQuaid, where the clergy's opinion was sought on every minor detail of society and state to one where it's seen as a gross impertinence for priests to have any opinions at all.
Maybe something valuable was lost in the process. It may seem like an amusing anachronism now to read a letter from three Aer Lingus hostesses in the Sixties -- on Gresham Hotel notepaper no less -- demanding that His Grace take action against a play running at the Gate Theatre which contained "open homosexuality" and "several scenes showing a most unhealthy attitude towards sex".
Likewise, to read another letter from a woman in Raheny denouncing actress Siobhan McKenna's performance in Brian Friel's The Loves Of Cass Maguire at the Abbey as "depraved".
"Her very posture is an insult to womanhood," the correspondent added. "The author, a Belfast-man [sic], must be an absolute gutter rat."
But at least they felt there was a sympathetic ear to share their disquiet at the direction in which the country was headed, whereas now they'd just be told to stop being so reactionary in case it upset any passing liberals.
As it happened, the archbishop could do little in either case, but it wasn't considered unreasonable for a Catholic prelate to have Catholic views and to express them vigorously.
That he was sometimes allowed to exert his influence rather more muscularly than we might now approve of is one thing; it goes against every contemporary instinct to see a middle-aged man in a cassock getting unduly exercised by the prospect of boys and girls playing games together.
But, then as now, it was up to politicians to set the proper limits of authority, and if they failed to do so satisfactorily then it's churlish to blame others for pushing at them.
Even more so now in a culture where everyone has an opinion and is encouraged to loudly let it all hang out. It just so happens that we do it on Twitter whereas John Charles McQuaid did it in pen and ink, but either way ours is the last society that should be chiding His Grace for not minding his own business.
'His Grace Is Displeased: Selected Correspondence Of John Charles McQuaid', edited by Clara Cullen and Margaret O hOgartaigh, is published by Irish Academic Press.