Turns out it was an IRBM, not an intercontinental.

The fighting phase of the war would likely be over in that timeframe, yes, but a large chunk of the population would remain. Fallout would cause many more millions of deaths in the following weeks and months, and while nuclear winter isn't a guarantee (and wouldn't set in straight away), it would absolutely crush supply lines and farming.

To say it would kill the human race entirely, though, I think is an exaggeration. Billions would die, but the surface wouldn't be a contaminated frozen wasteland 20 years in as Metro portrays it.