What was more responsible for the onset of war in the US the institution of slavery or the beliefs that states had the right to secede from the union?

It was the second one. The Confederates reasoned that the Union was a collection of states that had joined it voluntarily, and could secede equally voluntarily.

People like to claim that the war was caused by slavery. But they should ask themselves why Lincoln felt it necessary to issue the Emancipation Proclamation. In mid-war, he had to turn it into a crusade against slavery, to shame the British and French out of helping the South. That is the surest sign that slavery was not the original cause of the war.