So, in short anything which goes up to the supreme court would get stasised into an eternal limbo state because everyone can't meet? Or would it essentially be a referendum? That sounds pretty sensible, actually. If the courts can't decide something, it gets tossed to the public to decide?
To give an actually serious answer to this (as the failure to reach a decision does emerge in other scenarios as well), then you'd be left with the lower court ruling remaining intact. It would effectively nullify the Supreme Court as an entity, which would mean that there would be no more legal opinions binding the entire United States. (Sort of; court decisions relating to national agency actions would still have national ramifications.) Whether this is a good or bad outcome depends on one's philosophy, or just one's opinion regarding the trend of lower court opinions on a particular topic across jurisdictions.
One odd part of this is that it wouldn't be boiling down to different states, but a mix of federal appellate court jurisdictions - each of which are multi-state, and don't always make sense. E.g. Idaho would be stuck in the notably-liberal 9th circuit.
Edit: Wrote too quickly and forgot a few other implications. First, legal conflicts between states would no longer have an arbiter. So water rights disputes (often what leads to this) or border disputes wouldn't have formal resolution. Second, if I'm remembering right the federal court system would no longer have a way to override state supreme court decisions regarding federal matters - this is most relevant for situations where the federal constitution would override the outcome. For example, if [X state]'s supreme court holds that a racist law is fine, then normally the federal S. Ct. would step in and say 'nope, the federal constitution doesn't allow that'.
FURTHER EDIT: See a few posts down.
(I'm probably forgetting a few other situations.)
The other question that would get raised: if everyone was a justice of the supreme court, who would argue the case?