This is a provocative question, and it really depends on what you mean by a digital newspaper.
Arguably any written language formed out of a well-understood dictionary of symbols (like letters) could constitute a digital newspaper, since the signal distinguishing feature of "digital" technology is that a static discipline ensures that marginal inputs can become perfect outputs. This makes perfect copies possible and allowed scribes to preserve our culture for thousands of years, longer than the life of any physical medium.
Under this definition, the earliest handwritten gazettes, or even the texts of ancient civilizations could count.
But a major leap happened when perfect copies became not just possible, but easy and capable of mass production -- so maybe we should focus on the first typesetnewspapers.
Yet another leap happened when news began to be distributed not just in mass-produced digital form (the typeset broadsheet) that could be "easily" copied, but in electronic form that could be "instantly" copied.
Here we would be talking about the rise of telegraphy, which allowed the first stock tickers and the Associated Press and other newswires.
In the 60s and 70s, newspapers began to distribute national and international editions, sending their pages by satellite to printing presses across the country or in Europe. A battle ensued between the Wall Street Journal and the New York Times over the appropriate "digital" way to distribute their copy. The New York Times "digitized" its pages by scanning them and preparing a facsimile raster image, transmitting that pixel by pixel, and having it lithographed at printing presses across the country. The Wall Street Journal digitized its pages by coding the contents of each article letter-by-letter and sending that, along with a layout, to typesetters at each printing press who would have to re-typeset the pages before lithographing them. In John Hess's book "My Times," he describes how the NYT facsimile approach proved impractical with the technology of the day and was soundly beaten by the more data-frugal WSJ approach.
The 70s and 80s also saw the rise of electronic distribution to businesses and the public, with newspapers transmitting news copy to databases like Nexis and posting articles on online services like Compuserve and AOL. The 90s obviously saw extraordinary growth in the electronic distribution of news, as newspapers began posting their entire editions on the Internet.
I think if I had to pick one development along this timeline that signaled the birth of "digital" newspapers, it would probably be the invention of typesetting and the printing revolution, which made the mass-produced newspaper possible. It looks like that means I have to pick Carolus's "Relation," whose first edition was published in 1605.
But surely the invention of the telegraphic newswire is also a major, major advance that you could call the first "digital" newspaper.
And I think the anthropologists might point to the invention of written language in the first place -- and what it meant, namely the ability to perfectly copy texts and preserve them for longer than the life of any physical artifact -- as the signal development in our ability to use "digital" means to distribute and preserve our culture.