Without being very careful about how you regulate the use of 'public data' you could end up accidentally killing the internet.
With websites you can set a robot.txt file to opt out of indexing, and if they keep indexing anyway there are mechanisms to block it, both technical (i.e. ip restrictions) and legal (you can sue the indexer). The average website owners are also more technical and can understand the benefits and mechanisms, so presumably if you let your site gets indexed it's because you want to. And even so, what you want is for the search result page to drive traffic back to your site, not so that another site can be generated somewhere else and profited off by other.
With images, the artists are generally not technical and savvy enough to know how to opt out, there are currently no mechanism to opt out of indexing anyway once the image is uploaded to internet, the generated images do not increase awareness of the artist or drive traffic to their websites in any way (if they even have a website), plus the explicit goals of the resulting images are to make money for the model owner and the prompter.