Physicist: Probably not. We could kill all of the large (insects and up) life no problem. Hell, we’re doing all right by mistake so far. There are about 30,000 nuclear weapons in the world today, so in what follows I’ll assume the worst case scenario; that all of them are evenly spaced across the Earth’s land masses and set off. That should put them about 70km apart (in a grid).
Certainly everything on the surface within several dozen km of a nuke will be dead (like, really dead) but surprisingly, several feet of dirt or stone offer remarkable protection from the light and fire of the initial blast. Not directly under the explosion, but pretty close. It takes an amazing amount of energy to heat up and/or move dirt, so while the surface may be heated to red hot, the ground underneath can stay surprisingly cool.
So sure, you’ve kicked the legs out from under the ecosystem, but how do you ensure that you get everything? Fall out and nuclear winter are a good place to start. Nuclear winter is caused by dust thrown up in the air blocking out sunlight. The “sunlight blocking” shouldn’t last for more than a few weeks, but it takes very little time to starve all the plants and plankton that rely on sunlight. Or really just plankton, since you’re not going to find plants left standing within 35km of a nuke. Now, whatever survives (burrowing critters, seeds) will have to contend with ash instead of food, and radioactive fallout.
Modern weapons are fairly efficient, in that they use up almost all of their fissionable material when detonating. The initial flash involves a lot (as in “holy shit”) of radiation that mostly takes the form of gamma rays. Gamma rays are just high energy photons, so they’re gone immediately. Unfortunately, when fissionable stuff splits it breaks up into smaller isotopes which also tend to be highly radioactive. Most of these by products have short half lives. There’s a strong correlation between an isotope having a short half-life and the isotope radiating especially high energy crap when it decays. So most of the nasty stuff goes away pretty quick. The glaring exceptions to this are Caesium-137 and Strontium-90, which both have half-lives of about 30 years (and are delicious). Today the background radiation of Hiroshima is due primarily to Caesium, and that accounts for very little radiation total.
Basically, in order to survive the worst case scenario you have to: 1) live under ground or underwater, 2) be highly resistant to buckets of radiation, 3) not be particularly bothered by losing the sun for a while, and 4) not be particularly sad about the surface of the Earth burning and then freezing (or continuing to burn, just not as much. Some of the jury is still out).
We live in the largest ecosystem on the planet, but we definitely don’t live in the only one. There are fungus driven ecosystems deep in caves scattered around the world for example that may be safe. If however those caves can exchange air with the outside (or are forced to by a bomb for example), then the radiation would probably wipe out everything in there too. At the bottom of the ocean you can find black smokers, usually at the edge of tectonic plates. Black smokers are vents that spew out super-heated acid water laced with poison. I can only assume that the creatures that live down there must have been kicked out of every other clubhouse on the planet. These ecosystems depend only on heat and material from beneath the Earth’s crust, and as such are completely independent of the Sun. Although, poetically, since they depend on the nuclear decay of heavy metals in the Earth that were produced in at least one supernova more than 5 billion years ago, they still rely on a Sun, just not our Sun. The creatures in the black smoker ecosystems have to deal with radioactive crap flying out of the vents all the time, so they may be able to put up with fallout that manages to drift all the way down to them. Also, back in the 1950’s a bacterium called “Deinococcus radiodurans” was discovered that flourishes in radiation upto 3 million rads. By comparison, 1000 rads is usually fatal to people. 3,000,000 rads means that the glass of the test tube you’re keeping this bacteria in is going to turn purple and fall apart long before the bacteria dies.
I mean, how does that evolve? Where in the hell is this bacteria finding an environment that horrible?
Finally, Water Bears. God damn. Those guys don’t die. Ever. You can freeze them (-272°C), boil them (151°C), dry them out, irradiate them (500,000 rads), and even chuck them into space (seriously… space!), and they couldn’t care less.
So as long as there’s liquid water somewhere on Earth (even ultra-high pressure acid water) there will almost certainly be life. We would probably be more successful (at killing everything) with toxins and run-away global warming. So, if we could turn Earth into another Venus.
It worth noting that if this post seems a little “guessy”, it is. A lot of research has been done on the subject. The United States alone has detonated at least 1,054 weapons in tests, injected at least 18 people with plutonium, and exposed many more to radiation. The exact results of all these tests are largely classified (as in fact were the tests themselves). And of course, the world has never been destroyed by an all encompassing nuclear disaster. Hence the guess work.
However, we have fossil evidence of microbial life dating back about 3.8 billion years, and the moon’s marias were still being created (by really, really big impacts) until about 3 billion years ago. So we can expect that the Earth was subject to several ocean-boiling impact events since life started, and we’re still here (suck on that, space!).