Within JMP, once you have a prediction formula in a column then (no matter how that formula was created), adding a new row will give you a new, predicted value. So, to be clear, what you describe in the last paragraph is already possible in JMP.
But, as you imply, there are many use cases in which there is good reason to 'export' the JMP formula to another environment. To learn more about options for this, take a look at the Formula Depot.