There is a nice story:

Once in a physics class in Canada, the professor started to speak. "I can not see God, I can not touch God , therefore God does not exist"

A brave student , who sat in the back of the room, put up his hand to speak .

The Student said politely, "Professor, we can not see the Professor's Brain, we can not touch the Professor's
Brain, therefore The Professor has NO Brain."
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another story:

a Dreamer once asked a a sage "Who created the World?"
"God" replied the sage. The Dreamer demanded ,"Prove it!"
The sage only answered " Come back tomarrow, Please"

The Dreamer came back the next day . The Sage asked the Dreamer,"What are you wearing?"
The Dreamer replied," A handmade suit and pants"
The sage questioned, "Who made it?"
The dreamer responded," The Tailor"
The sage said," I don't believe you ! Prove it," demanded the Sage

The Dreamer said,"That is ridiculous. Can't you tell from the Fabric and the design that a tailor made this fine garment?"
"And you - YOU can NOT clearly tell that GOD made the World?" replied the sage
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The moral of these 2 stories is: Just as a house was obviously built by a builder and a garment obviously sewn by a tailor,
so is the World ,obviously made by a Creator called God

I don't think you would ever be able to 'prove' the existence of God based on an intuitive belief.

Isn't an intuition an inner certainty that is reached through personal reasoning, feelings and instinct?

Doesn't this rule out the possibility of proof? Surely, if the existence of God ever became provable the evidence would cease to be intuitive.

If it did become possible to prove the existence of God I suppose the ideas leading up to that evidence could be based on intuition - but not the proof itself, surely?