Great question, and short answer: yes, methods have already been developed and employed to grow artificial coral to fight against the loss of natural coral.
The longer answer is that the Note Marine Lab in Florida first noted that coral grows quicker when cut or fragmented due to its limited self healing properties, e.g. conditions have to be just right. They are one of the first to employ this idea on mass scale to grow coral artificially and transplant unto suffering coral reefs.
The group accomplishes this broadly by taking healthy coral of various species, cutting it into small pieces by a wet saw, then placing it on PVC pipe or cement/plaster frames shaped like trees in sea water to grow. Once the coral grows enough, some of it is reused in the growing process, while the rest is transplanted in groups and eventually fuses together as a replacement effort.
One last point of interest is that there are other techniques used to take advantage of the coral's self-healing, such as seeding the coral with small structures to enhance its regrowth.