I may have missed something, but the main quest says you only have 2 weeks to live. If I waste too much time doing side quests, will I fail the main quest?

Highly doubt it, it's not giving off that vibe.

I mean, I doubt it too, but it's weird that they give you this sense of urgency and then encourage you to spend days on side missions. I was just hoping there was a definitive answer somewhere.

Nope.

AFAIK it doesn't even keep track of the day, just the time of day.

Remember how in Witcher 3 the Wild Hunt was supposed to attack Kaer Morhen AT ANY MINUTE to get Ciri, but they were kind enough to wait until you gathered all allies scattered around the Continent, finished both expansions and gathered full decks of gwent cards? :)

So far I've only played only two games that had time limit in it's main plot that was actually real - Dead Rising 2 and Fallout 1 (there may be more, especially in older titles, I just don't know them). Now it's basically a trope for these kinds fo games where the world is almost destroyed or you're dying and need to hurry, but everything will always wait for you: Skyrim's Alduin, Oblivion's Daedric Invasion, Morrowind's Dagoth Ur's plan, Fallout 3's Enclave's plot (that's basically all Todd's games...), Dragon's Age invasion (semi-avoided because there are some quests that do move main story forward and lock your access to previous locations), Witcher 3's Wild Hunt (AFAIK, just saw some memes, haven't played it yet :D), Mass Effect's Reaper invasion, and so on and so forth. Kinda sad, since Fallout 1's system along with it's famous very expansive roleplaying branching makes it impossible to experience everything on one playthrough and makes you want to replay as another kind of character next time.