Well, spontaneous healing can't be proven scientifically. Its just a convenient way of getting around the paradox of a healing occurring which has no medical or scientific explanation.
There is no paradox - there will be a medical/scientific explanation: it is simply the case that the explanation is not yet fully known. Your focus on 'healing' is also misleading, since your are using it to imply the possibility of the divine in special cases only: those where the explanation is not yet available, which conveniently excludes all those where the explanation is understood.
The 'not yet known or fully understood' position doesn't just apply to unexpected recoveries of course, since at one time many of the conditions medical science has developed explanations for were similarly unknown. For example, the characteristics of Down's Syndrome was accurately described long before its cause (it is a chromosome abnormality) was known.
Seeing current ignorance as creating a gap for your god of choice is a fairly hopeless approach given that medical science hasn't stopped investigating.