Hey, I went ahead and checked a half-dozen sources for you; I went through Caesar's Gallic Wars, Civil War, Spanish War, Alexandrian War, and African War, Pliny's Natural History, Celsus' De Medicina, Suetonius, Plutarch, and even some of Cicero's works. Caesar, Suetonius (notorious for his gossip), and Plutarch (A Greek biographer) don't even mention dogs, while Celsus only touches on them briefly; he discusses how to treat dog bites here:

For almost every bite has in it poison of some sort. Therefore if the wound is severe, a cup should be applied straightway of it; if slighter a plaster, especially that of Diogenes. If that is not at hand, then one of the others I have recommended against bites; failing such, the green plaster called Alexandrian; if not even that is to be had, then any one which suits recent wounds, so long as it is not greasy. Salt is also a remedy for bites, especially dog-bite, if a hand is then placed over the bite and struck by two fingers of the other hand; for this brings out sanies; and brine-pickle may also be appropriately bandaged upon such a wound.

Celsus, however, seems to be rather grounded when it comes to medicine in general, so I turned to Pliny, who often writes down any and all rumours that he can get his hands on. For example:

[...] there are various other marvelous facts related, with reference to these animals. When a horse-shoe becomes detached from the hoof, as often is the case, if a person takes it up and puts it by, it will act as a remedy for hiccup the moment he calls to mind the spot where he has placed it. A wolf's liver, they say, is similar to a horse's hoof in appearance; and a horse, they tell us, if it follows in the track of a wolf, will burst asunder beneath its rider. The pastern-bones of swine have a certain tendency to promote discord, it is said. In cases of fire, if some of the dung can be brought away from the stalls, both sheep and oxen may be got out all the more easily, and will make no attempt to return. The flesh of a he-goat will lose its rank smell, if the animal has eaten barley-bread, or drunk an infusion of laser the day on which it was killed. Meat that has been salted while the moon was on the wane, will never be attacked by worms. In fact, so great has been the care taken to omit no possible researches, that a deaf hare, we find, will grow fat sooner than one that can hear!

Needless to say, he at the very least would have had something to say about the healing properties of dog spit. He talks about how to treat dog bites (far more fancifully than Celsus does - he suggests pressing veal to the bite will purge it), but interestingly enough, he doesn't mention any healing properties of dog saliva.

So I was a bit stumped. I haven't heard of those "healing properties" ever before either, yet this question suggests that the rumour was spread somehow. So I figured I'd buckle down a bit more. I searched Shakespeare's Julius Caesar. If anyone would make up something to make the man seem more fanciful, he's a pretty good candidate. Again, nothing.

The great news is that I DID eventually find something. The downside? There's no mention of Caesar. Grattius' Cynegeticon is a poem all about hunting, which as you can imagine, has a rather extensive discussion on dogs. When discussing how to treat wounds of those dogs, he says:

[...] if, however, there is merely a trivial hurt in a slight wound, the dog has the natural remedy of efficacious saliva.

That quote may have been as mistranslated as Grattius' name has been (Ending up as the rather hilarious "Gratius Falsius," which translates to "gratuitous hoax"), with authours quoting other authours' mistakes similarly to a game of telephone. I'm not sure of the exact origins to the myth about Caesar's "dog medics," but rest assured, his healers were human :)