Rarely do people break up after just one argument. Usually there are numbers of important underlying reasons that this relationship just isn't working. The argument immediately prior to breakup is usually the 'last straw' or an outburst of exasperation that the end has been reached which confirms those doubts that 'this isn't working' You can't change someone else. They are as they are, and if you don't like them as they are get away and out of that relationship. You should invest in a relationship that makes you happy, not one where it is full of disagreements, snipes and arguments.
You should also listen to your gut feelings here. A break up happens usually because one of the partnership is unhappy, and is a major sign the relationship will not work. Only you know whether your partner is a true match to you, and you both give equally and unconditionally to that relationship. Cut out the romantic stuff and ask whether you still want to wake up next to this guy in 20 years time. If the answer is no walk away.
He probably did want to hurt you if you have hurt him by rejecting him (and that pain is unavoidable) All he is doing is offloading his hurt back on to you. Don't prolong his agony by making contact or keep changing your mind (Why would you contact him again and then ask him if he was seeing someone else???) After you have broken the relationship he is perfectly entitled to see who he wants to. If you keep changing your mind it is unkind and destroys someone by giving them a sense of false hope. Some people have a very hard time managing rejection. It hurts their ego and they then try twice as hard to be so nice after that, but the outlook is still not a good one because nothing has changed in their core personality. Be sure of your decision and walk away. There are so many people in this world you haven't yet met. So much more life to learn about and a lot of years to go at without sticking to a relationship that is familiar but not fulfilling.
Walk away and find more about yourself and more about life for the next encounter.