Skin is a protective cover for our bodies. Once it is open, as mentioned above, it is a way for organisms to enter the body, thereby getting a potential infection. Depending on the person and their medical history, this could be a serious problem. Infections have become more serious these days compared to in the past. So the best solution is what Jemma suggested.
It may burn if some shampoo got into the cut but I doubt it would cause any permanent damage.
It would have likely caused an irritation or a severe skin reaction to one of the chemicals in that product.
As you had no adverse effect, then nothing will happen, but be very careful not to get anything on to open and broken skin. Broken skin is an entry for bacteria.
If you have a cut or broken skin due to spots, keep the area dry and avoid putting anything on it other than an antiseptic cream. Keeping it dry will help to heal it quicker. If you find a problem that it is not healing very well, get checked out by the doctor.
It probably would not do anything, but I would be careful for a few days until the cut is healed.