"To cut any ice" sounds somewhat old-fashioned to me. I'm having trouble figuring out what the sentence is supposed to mean, so I guess it doesn't sound terribly natural to me.
No, Johny, it doesn't mean that. 'Cuts no ice with' means 'doesn't make any impression on'.
'.
'When I took my little daughter's iPad off her she threw a tantrum, but it didn't cut any ice with me' (it didn't persuade me to give it back).