It is very unusual to get an infraspinatus injury with upright rows. Tendinopathy to supraspinatus and biceps tendons yes. That said there is a good load on the infraspinatus and teres minor during that exercise.

Perhaps he needs to watch you performing the exercise. Or better still get one of your friend to video you from behind (shirt off) and post it to youtube so we can take a look.

He is probably correct in that you scapula is not stabilising efficiently during the motion. As the infraspinatus comes off the scapula it requires it to be stable during the contraction. Thus working on exercise to stabilise the scapula is perhaps most important.

Overall one might also consider that you probably have an overactive left upper trapezius with heavier weights on that side. This can cause neck/nerve compression during the performance of the exercise and weakness may result. You may need complete retraining and even taping to inhibit your upper traps as part of the rehab process.

Perhaps a simple question, What are you doing that means you need to do so many upright rows? It is a slightly unnatural movement pattern when performed to the level of the shoulders.