A scribble on a screen is just as "worthless" as a scribble on a piece of paper in that anybody could have produced it. Even less so since all the nuances of a signature are lost in an on-screen signature and there are not even the diveshop stamps to back it up.
The only way to really "sign" a digital logbook is to use proper cryptographic-based digital signatures which would sign the important invariant parts of the dive data (date/time, location, depth, duration, buddy). This would still allow the diver to update and maintain the other data at a later time (eg, dive notes, equipment used) without invalidating the signature.
The technology to implement this is ancient. But the infrastructure doesn't exist, and would quite difficult to implement at the point of entry. Most people don't have a private key and those that do aren't in the habit of carrying it around with them. Although as smartphones become ubiquitous a simple app could solve those issues. It'd still need to be made simple to use though. And if verification really is the aim here, a dive shop would also need the relevant infrastructure in place to verify the signatures.
The alternative is online logbooks such as
diveboard which have a mechanism for buddies and dive shops to verify dives. The downside, of course, is that everybody has to have an account on whichever site the diver is using and the site has to establish some level of trust in its verification system.
Ultimately it seems there are only a few situations in which a logbook really does need to be verified and in that case it's probably best to fall back on the good old references system; as someone mentioned a few posts up, calling the shops and manually verifying a couple of random dives.
If dive verification becomes more important I'm sure the various apps will quickly offer solutions.