Can you breed new gold roses with existing gold roses on Animal Crossing New Horizons?

Golden roses won't clone or breed with another golden rose. According to the animal crossing fandom wiki:

While a golden rose cannot clone or breed with another golden rose, it can be used as a partner to another rose. For this purpose, it is considered to have a 'rryywwbb' genotype.

So yes, if you want more, you'll have to continue to water your black roses with the gold watering can.