Do modern-day Greeks feel continuity with their ancient civilization like Indians or Chinese?

They proclaim it and they are taught it, and yes, they feel it.

But they feel it at a superficial level, as either ancestor-worship, or a totem to beat up Westerners with. Nick Nicholas’ answer to If your country had a slogan what it would be?: “When we Greeks were building Parthenons, you barbarians were still eating acorns.”

If you dig deeper in getting how the Ancient Greeks ticked, you’ll see some superficial similarities, which Modern Greeks seize on—the fractiousness, the love of the good life, the politics. And you’ll see a lot more difference in how they viewed the world, and realise that there’s a reason why Etonians felt Demosthenes was their forebear rather than the Modern Greeks’.

But then, as I said in Nick Nicholas’ answer to Greeks, which do you identify most with: Ancient Greece or the Byzantine Empire?,

the marble looks like some highly advanced spacemen dropped this stuff off and left. It doesn’t gel with the Modern Greek landscape; it’s something Alien.

In fact, that’s how Modern Greek folklore accounted for all this marble. Built by the pagan giants of yore, before they collapsed under their own weight. And the Franks are the giants’ kinfolk; that’s why they come from their countries and genuflect before those ruins.

The theocracy of Byzantium really is more familiar, as Joachim Pense’s answer points out: “The continuity the modern Greeks feel mostly though, is that to the Byzantine tradition – of which the western Europeans don’t care much.”