Of course it depends on your definition of "classic", as that is a fairly subjective term, and it partially reflects whatever your personal taste in, and appreciation for, literature is. But since it was published in 2007, it is certainly contemporary.
I could add that as a winner of the Pulitzer Prize for Literature, perhaps it wins automatic entry into the classics club, although there are some past winners of that award rarely talked about or read today, so perhaps not. Personally, I think that The Road is not McCarthy's best work, (See Blood Meridian or No Country for Old Men as better examples of his talents), so in part I can see his receiving the Pulitzer as perhaps a reward for his lifetime achievements in literature.
I do believe both McCarthy's work and his reputation will stand the test of time, and we will one day refer to his books as classics with little debate, but I don't know if this book is there yet.
You would benefit from asking this question in the discussion postings to gain a wide range of responses, because this is one of those questions that really is very subjective. My short answer is that I do believe it is a contemporary classic because of the maturity of its style, theme and argument. I think as you read a book you can gain a sense of its maturity and its sophistication, and this novel scores highly on both in presenting a believable, realistic future world of brutality which really calls into question our identity and essence as humans as we are presented with a Darwinian dog-eat-dog existence that the boy desperately tries to raise himself out of and finds difficult to accept.
However, the only real way to assess whether a work is a "classic" is through the test of time, something which this novel has yet to achieve through no fault of its own. Unfortunately, there are so many good books that get talked about lots when they initially emerge into the publishing world, but are all but forgotten after a year or so. This novel is still being talked about, but we will see what happens in the next ten or twenty years.