Okay I figured it out! If anyone else has this error or problem, make sure you've actually downloaded the latest version of Minecraft. I hadn't actually run it on this computer yet. You have to load or play the vanilla, non-modded version of the version you are trying to install. For example, if you are installing the 1.16.5 version of Forge, you will need to have run the 1.16.5 version of vanilla Minecraft for it to work.
Hopefully this helps someone else!