Coming to a stop at a four signal. Light is green. I want to make a right hand turn. The person opposite me also is signaling to turn right. Who has the right of way?

You both do. If both of you are turning right and you are opposite each other, the both of you can move without impacting the other. The problem would occur if the other driver was making a LEFT, therefore, both of you would turning into the same street in the SAME direction.

My bad he is turning left. I'm turning right

Ahh, then you have the right of way.

Correct assuming you both arrived at the intersection at the same time and there are no arrows. In most states the laws tell you right of way is given not taken, always be cautious

I agree with the others on this. Keep in mind that as you turn right at a green light you don't typically come to a full stop (unless traffic or pedestrians on the side street force you to). But the fellow coming the other way who wants to turn left should be at a complete stop - he cannot proceed in case you decide to go straight through the intersection. As noted - this assumes that you both arrive at the intersection at the same time.