Can a graduate program rescind my offer if I don't complete a previously declared Minor?
I...declared a 'Minor in Computer Science (Expected)' on my US graduate school (PhD) applications. I have been accepted by a couple of programs...[C]ould [acceptance] be rescinded for not [achieving that qualification].


It depends on the offer. If an offer is conditional on the basis of you achieving a minor, then, yes, that offer can be rescinded. Otherwise (the offer is unconditional), it cannot. (Conditionality of an offer may be buried in the fine print.) Albeit, we're in legal territory and INAL.

For practical purposes, I suggest that you contact the universities that have made offers and discuss your situation with them.