As I understand it, this actually depends on how exactly the law is written. Some laws are written in so that their clauses are explicitly separate from each other. In that case, one clause may be struck down without affecting the others. However, if this explicit separation is not spelled out, then the whole law must generally be taken as a single entity, that fails entirely if any part of it fails.
It may also be the case that the clauses of the law must be linked in order for the law to succeed. This is the kind of thing that courts must rule on. Another thing that courts may have to rule on is any custom or precedent that might apply to a specific category of law, where the laws are always assumed to be severable (several?) even when not spelled out.
All of which to say that it's not automatically the case with every law, that striking down one clause doesn't mean striking down the whole law.
I seem to recall some discussion a few years ago about how the ACA omitted the explicit wording that allowed its clauses to be ruled on separately from each other. But I don't have any details and can't vouch for the accuracy of that.