I traded in my copy of overwatch and deleted the game. Would I be able to reinstall it from the library?
No, this is not possible.
Games must have been purchased/downloaded from the PS Store in order to be installed via the library. Physical versions aren't tied to your account in the same way as digital, otherwise I could just sell all of my physical games immediately and redownload them digitally, allowing me to play for practically no cost.
You sold the physical game and therefore no longer own a copy. You must buy a new one. This is unlike digital games, where you obviously cannot sell away your copy, so you are free to redownload it whenever - you still "own" it (or rather, you're purchasing a license to use it, following Sony's Terms and Conditions).
Assuming you've actually deleted all data associated with the physical copy, you shouldn't run into any issues installing/playing the digital version.
In addition to @BrootsWaymb's answer, even if you didn't delete the game (or managed to download it again), you wouldn't be able to play it.
Physical copies of PS4 games can only be launched with the disk still in the system. The PS4 won't let you play otherwise.
The only options you have at this point are:
- Buy another physical copy and avoid "losing" the disk. You can reinstall the game at will, as long as the disk's in your PS4's disk drive and still in readable condition. You'd need the disk there anyway, if you intend to play the game.
- Buy a digital copy. You can re-download the game at will, unless Sony removes it from their servers (like what happened to P.T.). Those situations are extremely rare, though they do happen occasionally. You gain the benefit of not having to swap disks, in return for not being able to resell them.
Of course, there's always the option of simply not playing the game anymore, since you sold it.
Note that the game data are different from the save data. If you deleted the game data, you can easily recover those (as described above). If you delete your save data, on the other hand, there's no going back. You might still have a copy in the cloud if you are a PS Plus member and have the auto-upload option enabled, unless you deleted that one as well, in which case have fun starting the game anew!