I think that there are several. The more of them you have, the closer you get, in my opinion.
First is reverence for Celtic deities. This is easy, and pretty widespread, even among groups that are not really Celtic in focus. Lots of purely Wiccan groups, for instance, revere Celtic Gods and Goddesses, without fulfilling any of the other possible criteria.
Second, connection with ancestors and land spirits. This one is pretty generic and needs to be taken in combination with several other things, because ancestor worship and reverence for land spirits happens in most old Pagan cultures. I would suggest that this connection and reverence must happen in a style not unlike that shown in Evans-Wentz's "The Fairy Faith in Celtic Countries" for it to be seen as a continuation of the Celtic spirit. We can carry it forward into a modern Celtic spirit by having a general love and reverence for the earth and its creatures. A deep appreciation of nature is revealed in early Celtic nature poetry from Ireland and Wales.
Third, poetry as intrinsic to the structure of magick. Lorax and I have done a number of rants on poetry here. We're not talking about lame moon/June/tune rhymes, but about the kind of poetry that stirs up fire in the soul, the kind that speaks power in its descriptions and its focus. The sort of poetry that sucks you in and churns your guts. Although we often get clinical in our writing, we also try hard to make much of our writing lyrical in that sense. I hope that we sometimes succeed. In addition to poetry as magick, there was also respect for poetry as a social mechanism; it offered praise for those who were worthy, and satire and scorn for those who were not. It isn't just the reading of poetry, but the making of poetry that is important. Celtic Pagans must be poets, even if they aren't great poets.
Fourth, a connection with the past. The Celts had a reverence for history, and that reverence is a part of the Celtic spirit, I believe. For some, this connection comes through physical ancestry. For others, it comes through study of history. Some people get it through connecting with the feeling of the myths. Other folks get it in other ways. I think that this is why we have such heated debates here about the importance (or lack thereof) of sticking to historical fact. We all recognize that something from the past is speaking to us strongly, but we disagree about the methods of judging its veracity and usefulness.
Fifth, a sense of early Celtic cosmology; doing things in terms of three realms rather than the classical Greek four elements, using Celtic symbols like triskeles and spirals rather than pentagrams, celebrating Celtic holidays rather than (or more deeply than) the holidays of other religions, threes and nines as ritually important, use of a sacred/cosmic tree and well combination. Much of this cosmology has had to be painstakingly reconstructed from fragmentary hints, and it goes back again to the argument that historical research is important to learning about and preserving the Celtic spirit.
Sixth, I think that inclusiveness is important. We can't rely on genealogy or geography to determine who is Celtic. The historical Celts roamed all over Europe, and lands beyond. Anyone worthy might be taken into the tribe through marriage or adoption. The Celts are roaming still, moving to America, Australia, and other widely diverse lands. And they're still taking people in through marriage and adoption.
Seventh, respect for women was a definite part of the Celtic spirit. While Celtic women didn't have it perfect, they were far better off than their Greek and Roman counterparts. Likewise, respect for and acceptance of gays and lesbians seems important. There is certainly text evidence for men loving men in early Celtic society. Women were not as often written about, but I think it is safe to assume that women had similar choices open to them.
Eighth, an appreciation of the complex and intricate. This is found in Celtic art, law, myth and poetry. The classical historians noted that the Celts spoke in riddles and loved to obfuscate. Wordplay and veiled reference were common.
Ninth, personal responsibility and a deep sense of self are a part of the Celtic spirit. Boasting and personal pride are evident in every Celtic tale. Sometimes it went overboard, so of course, like some other things (head hunting, etc), we have to be careful not to get too deeply into it. I think that some of us do act on this Celtic instinct, and that's why we often have heated debate on this list. So long as it doesn't get out of hand, I find it encouraging and a growth-oriented activity. Spirited argument was a part of the poet's duty, and was one of the ways in which the younger poets learned from the older. Along with this, I would say that the Celtic spirit includes a strong sense of ethics about what is right and what is wrong. The Celts were not an "anything goes" kind of people. They had a very complex body of laws governing what was appropriate and what was not. Celtic Pagans need both a strong sense of personal responsibility and a code of personal and social ethics in order to carry the Celtic spirit forward.