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Wikipedia has a sentence in its article on ellipsis:

In reported speech, the ellipsis is sometimes used to represent an intentional silence, perhaps indicating irritation, dismay, shock or disgust. This usage is more common amongst younger, Internet-savvy generations.[citation needed]

I can find plenty of random internet articles making a similar statement, but is there an actual authoritative grammar source that says this is grammatically correct? Is it just something the "Internet-savvy generations" have invented?

Here is an example from a story:

She swallowed hard. "I'm afraid, Mark. Maybe if you might... talk to him?"

Here the ellipsis is indicating a verbal hesitation in the quote rather than the typical use of indicating an omission.

According to Grammar Girl, several style guides support the use of ellipses to indicate a pause (the relevant paragraph can be found under the header The E-mail Ellipsis).

She quotes from the Chicago Manual of Style that "Ellipsis points suggest faltering or fragmented speech accompanied by confusion, insecurity, distress, or uncertainty."

I would consider such style guides to be the kind of authoritative source you were looking for.

I'm far from convinced there's any justification for that Wikipedia claim about this usage being more common amongst younger, Internet-savvy generations, except in the sense that many young people today probably actually write more often than earlier generations anyway, because they "chat" using text on social network sites, and post on forums. So you could say every aspect of writing is more common among the younger generation. But so far as I know, it's always been a common device, so the supposed "Internet" connection is spurious.

As I write, I see Bjorn has just posted this same link to Grammar Girl, which cites CMOS if GG isn't authoritative enough for OP (I think she's fine in her own right, though I don't always endorse CMOS "recommendations").

It's worth noting that in general, CMOS is "prescriptive" - it's a style guide, which specifies a particular (hopefully consistent) set of guidelines. Grammar Girl is primarily "descriptive" of all different usages which do in fact occur, without necessarily promoting one over another.

The question asks whether this use of punctuational ellipsis is "grammatically correct".

However, this is not a question of grammar at all. It is rather an issue dealing with English writing, not with the English language per se. Writing is just technology; it's (real, spoken) language that has grammar. English punctuation, in particular, is not fixed by any agreed-upon rules, but rather is employed quite variously, as it always has been.

The fact that (real, actual spoken) English grammar is not taught in Anglophone schools has resulted in people using the word grammar to refer to just about anything they were supposed to learn at school about writing that they're unsure on -- including punctuation -- and also to use the term "grammatically correct" to refer to the imagined solution to their puzzlement about it.

I would suggest that, if a writer believes their reader(s) will understand their phonotactic intention in using an ellipsis, then the writer should go ahead and use it. "Correctness" is of no consequence here; effectiveness is.

For instance, I was puzzling the other day how to represent the intonation I wanted in the mind's ear of a reader, with a post that started out: Well ... I guess you could say that. I fussed with commas and other things and finally wound up using the ellipsis to indicate that the intonation slopes down for a while till the I is pronounced.

Note, this wasn't a pause; it was a longer-than-average well, with the final lateral resonant stretched out on a downward intonation contour. Any English speaker knows what that sounds like and what it means; the problem is representing it effectively in writing, which is preferable to smoke signals for representing English intonation and rhythm, but only just.

Larry Trask limits the use of ellipsis to showing 'that some material has been omitted from the middle of a direct quotation' and to showing 'that a sentence has been left unfinished'. There is thus the possibility that the ellipsis will be ambiguous if it is also used to indicate a pause, but I can see no other way of doing so.

What I see in style guides regarding the usage of the ellipsis for an omission is presented as three periods separated by spaces (. . .) for the ommission. The ellipsis without the spaces (...) for the pause.

Any author is perfectly within her rights to tell the punctuation police to **** off.

These are matters of convention, not grammar, especially in fiction, and authors of fiction have license to approach conventions as they see fit.