Why do Paris and Romeo fight even though Romeo really has no wish to fight anyone?

He was devastated by the fact that Juliet had died. At her tomb, Romeo was going to kill himself when Paris showed up. Romeo wanted to be alone and when Paris would not leave he challenged him. Romeo did not want to kill him but he had to.

It's Paris that starts the fight. He assumes that Romeo is there "to do some villainous shame to the dead bodies". As usual, he pays no attention to what Romeo is telling him, and intends to "apprehend thee for a felon here", that is, to arrest Romeo for being in Verona when he is banished. Romeo has tried his best to explain, saying "I love thee better than myself, for I come hither armed against myself." and explaining that he planned to die there, which would have made some impression on anyone other than thick-as-a-brick Paris.