Is it possible to install already installed apps with one click?
This doesn't specifically exist, however there is the Android Backup System, which allows you to reinstall all the apps from the Google server's on the inital setup of the device.

This will queue all the apps for install from Google Play. It comes on all Android 2.2+ devices, however it is very temperamental.

Other than that, you could use a dedicated backup app such as Titanium Backup Pro to regularly backup your apps, allowing you to restore them in one click should you need to.

Google Play doesn't offer what you stated. iOS offer's this because they don't have permissions to show you ;)