To which I can only say, good question. Veyne's answer is No, But.
And herein lies the problem: while this is a slim book (129 pages, plus another 22 pages of small-font notes), without the repetition and meandering it cd have been slimmer still: I think Veyne cd have said everything he had to say worth saying in twenty pages or so. Here are a few of his main points:
(1) the Ancients thought mythical time had been different from the contemporary time they themselves lived in. So there might have been monsters in the time of Hercules or Odysseus, but not anymore. This reminded me v. much of Kordecki's dissertation, which concluded that folks in the Middle Ages believed in dragons because they had so much evidence (in the form of old stories, including multiple mentions in the Bible, now re-translated away today) that dragons had once existed. But they didn't think they still existed as something you could run into 'nowadays', in their equivalent of modern times. Similarly, I know of some Xian denominations that believe the Age of Miracles ceased with the death of the last of Christ's original disciples, John, at Patmos around 100 AD. Veyne draws the demarcation line as about the time of the Trojan War, after which Gods ceased to appear and epic monsters died out.
(2) Disbelief took the form not of rejecting myths but of trying to rationalize them. For example, by late Hellenic/early Roman Empire times writers and thinkers didn't believe in the Minotaur but instead thought it'd been a person named Taurus who held an important post under Minos (pretty much the solution Mary Renault came up with in her Theseus novel). They didn't doubt that there had once been a king named Minos, just that the supernatural stories connected with him were exaggerations beneath which lay historical facts, recoverable to the sharp-witted. I was reminded of people who try to prove scientifically that the Star Over Bethlehem was some sort of nova or Velikovsky's account of the Parting of the Red Sea. As I understand it (second-hand, never having read Velikovsky), he never doubted that the sea opened up and let Moses and the Israelites pass, then came rushing back and drowned the Egyptians, but instead of a miracle thought it had been caused by a catastrophic planetary alignment (Venus passing too close to the Earth), which occurred again a generation or so later, causing the sun to stand still in the sky (that is, the Earth to stop rotating for a few hours). A modern skeptic wd simply doubt that the Red Sea miracle occurred at all; a non-literalist Xian wd be closer to Veyne's Romans and latter-day Greeks, believing in the people but not the specific events.
Perhaps Veyne's most interesting point is that the Greeks (and Romans) simply cdn't conceive it was all just made up. There must have been a Romulus, a Theseus, and so forth. I'm told there's exactly as much archeological evidence for the existence of King David and Solomon as there is for King Arthur (i.e., none), so perhaps we're not so v. different in what we choose to believe. His passage on the psychology motivating "sincere forgers" -- i.e., folks who at some point made up detailed geneologies out of whole cloth -- is also interesting.
Unfortunately, Veyne's methodology is somewhat suspect. For example, when someone like Aristotle introduces a reference to a myth with "it is said" or some such phrase, Veyne asserts that this means Aristotle is revealing that he doesn't believe a word of it. Well, maybe. But maybe not. It's too subjective a claim to settle such a fundamental point essential to his argument.
Oddly enough, I'd recommend skimming this book and then reading the Endnotes, which are wonderfully detailed and much better written than the main text, which loves to say and unsay and re-say and assert and retract and generally mull out loud over the same points time and again.
So, a fascinating topic, and it'll make you think, but in the end it doesn't really satisfactorily answer its own question.