It could, depending on how it was executed, on what scale, and such.
Since you and your buddy agreed on the word "holocaust", I'll assume for the sake of argument that we're not talking about *one* isolated explosion. One or two of those at a time are survivable, on the scale of the rest of the human race, as anyone who knows of the detonations at Hiroshima and Nagasaki can tell you.
But on the other end of the scale, you have Mutually Assured Destruction (MAD), which was the scenario that played out between the United States (and NATO allies) and the former Soviet Union (and former Warsaw Pact allies). Basically what that amounted to was all-out warfare involving *hundreds to thousands* of nuclear bombs and missiles launched, at every *major population center* on both sides.
Something on *that* scale could make us extinct, especially if many of those bombs detonated *on impact with the ground* instead of at low altitude over their targets. The difference? A ground-zero detonation *kicks up* a lot of radioactive debris, called "fallout", which then gets carried by local winds (think of the jetstream, high-altitude stuff), spreading radiation and its poisoning around.
Where, with a low-altitude detonation, you get more *damage* at the target, but way less fallout kicked up. The heat and radiation are more contained at the target.
But yeah...if a lot of the bombs and missiles in a MAD scenario actually hit the ground and then blew up, there would be enough fallout in the air to *sterilize* whole continents, or to at least render them as poisonous as Chernobyl.
The catch? The overwhelming majority of these targets lie *North* of the planetary Equator. One could argue that large chunks of the South American continent (well south of the Panama Canal, never mind Mexico), as well as chunks of Africa, Indonesia and *all* of Australia would be left out.
But that's where the fallout comes in. Eventually rivers would be poisoned, which means in time *oceans* would be poisoned. And this doesn't even count any El Nino/La Nina patterns changing the jetstream, or any "nuclear winter" scenarios arising from fallout in the upper atmosphere *blocking out the Sun*.
Eventually, the oceans would be poisoned enough by fallout to get to Australia and the most remote Pacific Islands. It would take time, but it would happen with a high-fallout nuclear war.
Humanity would be extinct *immediately* North of the planetary Equator....and it would be extinct South of the Equator within 20 years or less.
But don't just take my word for it. Search the site below for the whole phrases "November 2007" and "Nuclear Weapons in a New World" (which is the name of a recent article):
http://www.sciam.com/
That should tell you much of what you need to know.
Thanks for your time.