Based on the http://www.nzta.govt.nz/resources/roadcode/about-driving/merging.html page the car on the right should have adjusted speed to merge. Yes it was on the right however it was merging with the main flow of traffic therefore it was at an intersection. All the OP's car should have done was keep speed and not speed up or slow down and allowed the car on the right to merge safely.
Leaving a merge lane after turning right at an intersection
Some roads have a merge lane to help vehicles that have turned onto the road join the traffic flow.
If you are in a merge lane at an intersection:
- show you want to merge by signalling for at least three seconds
- move into a safe gap in the traffic
- adjust your speed and following distance.