Should abortion be left to the states?

Many people -- some for abortion rights and some against abortion rights -- think that abortion laws should be decided by each state, rather than dictated by the federal government. People mostly argue for that view on the grounds that the right to abortion is not mentioned in the Constitution. I've also seen someone argue that since murder is defined by state law, whether abortions are deemed murder or not should be a matter of state law. What would result from allowing states to decide abortion law? Would that be the right approach?