But is there any evidence that would persuade a nonbeliever that God was behind it — that we do not live in a closed system in which all causation is a matter of natural laws?
No. Jesus said so explicitly in the parable of Dives (the rich man) and Lazarus:
“He said to him, ‘…they will not be convinced even if someone rises from the dead.’”
Luke 16:21
Miracles can happen too without breaking any natural laws, only spectacular timing and placing – even Jesus calming the storm on the Sea of Galilee was one of those. A man in a boat said something (to the storm!) and it became ‘completely calm’.1 What natural laws were broken? Cause and effect could not be ‘proven’.
Special providence is like that too, as evidence demonstrates, although some foolishly say that it is laughable that God is a loving Father who is sovereign over time and space, timing and placing. My nephrectomy does not demonstrate miraculous healing, but it most certainly does demonstrate God’s providence (especially for my wife, since I don’t want to leave her here without me, not that I’m not ready or that God wouldn’t take care of her, but it would be tough on her! ; - ).
1 (And they had to row… all the rest of the way, demonstrating that faith and trust can be easier than not. ; - )