"Now if there was ANY case of a successful supernatural explanation of anything in the natural world perhaps you might have an argument as to the existence of God. But there isn't."
Any religion with a true believer satisfies your premise. If you are looking for supernatural things that have been subjected to modern scientific scrutiny, read up on:
-the Miracle of Lanciano (I've seen this first hand-no apparent change last year since the earliest pictures)
-the Shroud of Turin
Note that the Shroud has not been proven to be a medieval forgery as often cited; read up on the follow-up study by the original dating team which found the original carbon dating to be invalid because it incorporated fibers from a medieval repair effort. Don't limit yourself to reading the summaries; the case for an invalid dating is rock solid as stated by the man who led the initial dating effort (the original dated samples contained a mixture of linen fibers from the original fabric and died wool from the repair which resulted in a date weighted significantly in the direction of the repair date due to exponential decay).
Also, there are many examples of "miraculous" occurrences in modern settings that are well documented, but ignored or unknown by nonbelievers. If you don't think any of them are credible, take it upon yourself to talk to a number of medical doctors (especially ER) and ask how many have seen things that they contradict everything they know about medicine; the likelihood of finding something that has yet to be explained by science is pretty high in any population of doctors or emergency workers.