The "I'm not a robot" captcha (checkbox) will validate once the box is checked. So it will be validated prior to the form being submitted. Secondly, if the form is submitted and the captcha has been invalidated or is invalid, the form submission will fail. So no need to perform any further validation as this is already done for you.
Have you looked at the New Recaptcha project code at all? I say download the code and read the documentation and make modifications as you see fit in the project code.