1)
Do you have an R folder located in your Documents? This folder should contain the win-library folder and is where the installer will initially look to install the arcgisbinding package to.
Yes, I have two R folders:
My Document/R/win-library/3.3
My Document/R/win-library/3.1
2)
Have you modified any of your environmental variables or path regarding R?
No, I didn't find any variable that looks R-related, and don't have R_HOME.
By the way, I should have specified earlier that I'm under Windows 7.
3)
What happens when you try to install the bridge from file?
1. I downloaded arcgisbinding_1.0.0.128.zip
2. In Options/Geoprocessing/R-ArcGIS Support I selected "Install from file"
3. I navigated to where I had saved the file, and selected the file
4. There was briefly (1second) a blue status bar showing installation until the same pop up error appeared ("'C:\Program' is not recognized as an internal or external command, operable program or batch file").
Here's more info on my machine and my R version
Machine
Windows 7 professional
Service pack 1
Intel (R) code (TM) i5-6300U CPU @ 2.40GHz
8Gb RAM
64-bit operating system
R