The installer seems to create a version JSON that is incorrect, or at least that the launcher doesn't like.
Steps:
Move current .minecraft directory out of the way.
Launch the official Minecraft launcher, log in, launch the game once and quit.
Grab https://files.minecraftforge.net/maven/net/minecraftforge/forge/1.13.2-25.0.26/forge-1.13.2-25.0.26-installer.jar and run it.
Notice the installer says it installed successfully.
Launch the Minecraft launcher again.
Notice that forge doesn't launch and when you edit the profile the version of forge is not there (even though when you look at the profile list it looks right).
Notice the crash log.
Crash log:
Workaround:
Edit the version json to put "http://files.minecraftforge.net/maven/" in the url field for forge (corresponding to source code line https://github.com/MinecraftForge/MinecraftForge/blob/1.13.x/build.gradle#L492)
With the file saved, re-launch the launcher.
Play modded minecraft.