There’s no need to prove God’s existence to Hawking or anyone else. He and they will eventually find out for themselves. That is why He says those who do not believe in Him are fools.
I pray that men like Hawkings see the truth sometime prior to their earthly demise; repent; and accept Jesus as their Lord. Otherwise, he will be pontificating throughout eternity on questions like: Will time end? and How long can Hell burn before its source of fuel is exhausted?