Is there any Magento extension that works like PayPal Pro website payments? I am looking for on site credit card processing using Google Checkout.
I am certain Google Checkout isn't meant to be completed on your website. It will always require the user to leave to go to the Google Checkout page.
Looking at the overview of Google Checkout it says: Buyers' credit card details are given directly to Google, so even if they're unfamiliar with your store, they'll be able to trust that their information is secure — while you won't have to worry about safeguarding it.
They also go on with a simple diagram explaining the process.
Looking at a comparison between PayPal & Google Checkout, they are also saying it can't be done on site. http://paypalvsgooglecheckout.com/compare-paypal-google-checkout/
The Google Checkout option is currently installed in the base code of Magento. It is located under "Google API" in the backend.
You should be able to use Google checkout but it will require you to create a custom module with a lot of API work.
I would advise you against using Google Checkout as they are retiring the product in november this year.